PD13 AbstractsPD13 Abstracts 57 [email protected] IP5 Tug-of-war and Infinity Laplacian with Neumann...
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56 PD13 Abstracts IP0 The SIAG/Analysis of Partial Differential Equa- tions Prize Lecture: Weak Solutions of the Euler Equations: Non-Uniqueness and Dissipation There are two aspects of weak solutions of the incompress- ible Euler equations which are strikingly different to the behaviour of classical solutions. Weak solutions are not unique in general and do not have to conserve the en- ergy. Although the relationship between these two aspects is not clear, both seem to be in vague analogy with Gro- movs h-principle. In the talk I will explore this analogy in light of recent results concerning both the non-uniqueness, the search for selection criteria, as well as the dissipation anomaly and the conjecture of Onsager. L´ aszl´oSz´ ekelyhidi, Jr. Universit¨atLeipzig [email protected] Camillo De Lellis Institut f¨ ur Mathematik Universitat Zurich, Switzerland [email protected] IP1 Coupling Between Internal and Surface Waves in a Two-layers Fluid Internal waves occur within a fluid that is stratifed by tem- perature or salinity variation. They are commonlygener- atedinthe oceans. They havetheform of large-amplitude, long-wavelength nonlinear waves that propagate over large distances. In some physically realistic situations, internal waves give rise to characteristic features on the surface, a signature of their presence, in the form of narrow bands of rough water, sometimes referred to as a rip, which prop- agates at the same velocity as the internal wave, followed after its passage, by the complete calmness of the sea, the mill pond effect. Our starting point is the two or three- dimensional Euler equations for an incompressible, irrota- tional fluid composed of two immiscible layers of different densities. We propose an asymptotic analysis in a scaling regime chosen to capture the observations described above. The analysis of the asymptotic model shows that the rip region of the free surface is generated by the resonant cou- pling between an internal soliton and the free-surface wave modes while the mill pond effect is the result of a domi- nant reflection coeffcient for free-surface waves in a frame of reference moving with the internal soliton Catherine Sulem University of Toronto [email protected] IP2 Partial Regularity for Monge-Ampre Type Equa- tions Monge-Ampre type equations arise in several problems from analysis and geometry, and understanding their regu- larity is an important question. In particular, this kind of equations arises in the regularity theory of optimal trans- port maps. In the 90’s Caffarelli developed a regularity the- ory on R n for the classical Monge-Ampre equation, which was then extended by Ma-Trudinger-Wang and Loeper to a more general class of equations which satisfy a suitable structural condition. Unfortunately, this condition is very restrictive and it is satisfied only in very particular cases. Hence the need to develop a partial regularity theory: is it true that solutions are always smooth outside a ”small” singular set? The aim of this talk is first to review the classical regularity theory, and then to describe some re- cent results about partial regularity. Alessio Figalli Department of Mathematics The University of Texas at Austin fi[email protected] IP3 Waves in Honeycomb Structures I will discuss the propagation of waves in honeycomb- structured media. The (Floquet-Bloch) dispersion rela- tions of such structures have conical singularities which occur at the intersections of spectral bands for high- symmetry quasi-momenta. These conical singularities, also called Dirac points or diabolical points, are central to the remarkable electronic properties of graphene and the light- propagation properties in honeycomb structured dielectric media. Examples of such properties are: quasi-particles which behave as massless Dirac Fermions, tunability be- tween conducting and insulating states and topologically protected edge states. Most theoretical work on honey- comb structures (going back to 1947) has centered on the tight-binding approximation, a solvable discrete limit cor- responding to infinite medium contrast. We present re- sults (with CL Fefferman) for the Schroedinger equation with a honeycomb lattice potential with no assumptions on medium contrast showing: a) the existence of conical singularities in dispersion surfaces for generic honeycomb lattice potentials b) the persistence of such conical singu- larities under perturbations which preserve time-reversal and spatial inversion symmetries, e.g. a linear strain of the honeycomb structure c) that wave-packet initial condi- tions, which are spectrally localized about a Dirac point are governed, on very long time-scales, by a system of homogeneous 2-dimensional Dirac equations. Finally, we discuss the question of topologically protected edge states and recent work in this direction (with CL Fefferman and J. Lee-Thorpe). Michael I. Weinstein Columbia University Dept of Applied Physics & Applied Math [email protected] IP4 Granular Experiments of Discrete Nonlinear Sys- tems Ordered and disordered arrangements of granular particles are governed by highly nonlinear contact interactions that confer to the particles assembly unique dynamic properties. We study how stress waves propagate in lattices composed of particles in close contact, and exploit this understand- ing to create novel materials and devices at different scales (for example, for application in energy conversion, vibra- tion absorption and acoustic rectification). We control the constitutive behavior of these new materials selecting the particles geometry, their arrangement and materials prop- erties. We assemble and test experimental systems and in- form our experiments with discrete numerical simulations. Chiara Daraio ETH Z¨ urich
Transcript of PD13 AbstractsPD13 Abstracts 57 [email protected] IP5 Tug-of-war and Infinity Laplacian with Neumann...
56 PD13 Abstracts
IP0
The SIAG/Analysis of Partial Differential Equa-tions Prize Lecture: Weak Solutions of the EulerEquations: Non-Uniqueness and Dissipation
There are two aspects of weak solutions of the incompress-ible Euler equations which are strikingly different to thebehaviour of classical solutions. Weak solutions are notunique in general and do not have to conserve the en-ergy. Although the relationship between these two aspectsis not clear, both seem to be in vague analogy with Gro-movs h-principle. In the talk I will explore this analogy inlight of recent results concerning both the non-uniqueness,the search for selection criteria, as well as the dissipationanomaly and the conjecture of Onsager.
Laszlo Szekelyhidi, Jr.Universitat [email protected]
Camillo De LellisInstitut fur MathematikUniversitat Zurich, [email protected]
IP1
Coupling Between Internal and Surface Waves in aTwo-layers Fluid
Internal waves occur within a fluid that is stratifed by tem-perature or salinity variation. They are commonlygener-atedinthe oceans. They havetheform of large-amplitude,long-wavelength nonlinear waves that propagate over largedistances. In some physically realistic situations, internalwaves give rise to characteristic features on the surface, asignature of their presence, in the form of narrow bands ofrough water, sometimes referred to as a rip, which prop-agates at the same velocity as the internal wave, followedafter its passage, by the complete calmness of the sea, themill pond effect. Our starting point is the two or three-dimensional Euler equations for an incompressible, irrota-tional fluid composed of two immiscible layers of differentdensities. We propose an asymptotic analysis in a scalingregime chosen to capture the observations described above.The analysis of the asymptotic model shows that the ripregion of the free surface is generated by the resonant cou-pling between an internal soliton and the free-surface wavemodes while the mill pond effect is the result of a domi-nant reflection coeffcient for free-surface waves in a frameof reference moving with the internal soliton
Catherine SulemUniversity of [email protected]
IP2
Partial Regularity for Monge-Ampre Type Equa-tions
Monge-Ampre type equations arise in several problemsfrom analysis and geometry, and understanding their regu-larity is an important question. In particular, this kind ofequations arises in the regularity theory of optimal trans-port maps. In the 90’s Caffarelli developed a regularity the-ory on Rn for the classical Monge-Ampre equation, whichwas then extended by Ma-Trudinger-Wang and Loeper toa more general class of equations which satisfy a suitablestructural condition. Unfortunately, this condition is veryrestrictive and it is satisfied only in very particular cases.
Hence the need to develop a partial regularity theory: isit true that solutions are always smooth outside a ”small”singular set? The aim of this talk is first to review theclassical regularity theory, and then to describe some re-cent results about partial regularity.
Alessio FigalliDepartment of MathematicsThe University of Texas at [email protected]
IP3
Waves in Honeycomb Structures
I will discuss the propagation of waves in honeycomb-structured media. The (Floquet-Bloch) dispersion rela-tions of such structures have conical singularities whichoccur at the intersections of spectral bands for high-symmetry quasi-momenta. These conical singularities, alsocalled Dirac points or diabolical points, are central to theremarkable electronic properties of graphene and the light-propagation properties in honeycomb structured dielectricmedia. Examples of such properties are: quasi-particleswhich behave as massless Dirac Fermions, tunability be-tween conducting and insulating states and topologicallyprotected edge states. Most theoretical work on honey-comb structures (going back to 1947) has centered on thetight-binding approximation, a solvable discrete limit cor-responding to infinite medium contrast. We present re-sults (with CL Fefferman) for the Schroedinger equationwith a honeycomb lattice potential with no assumptionson medium contrast showing: a) the existence of conicalsingularities in dispersion surfaces for generic honeycomblattice potentials b) the persistence of such conical singu-larities under perturbations which preserve time-reversaland spatial inversion symmetries, e.g. a linear strain ofthe honeycomb structure c) that wave-packet initial condi-tions, which are spectrally localized about a Dirac pointare governed, on very long time-scales, by a system ofhomogeneous 2-dimensional Dirac equations. Finally, wediscuss the question of topologically protected edge statesand recent work in this direction (with CL Fefferman andJ. Lee-Thorpe).
Michael I. WeinsteinColumbia UniversityDept of Applied Physics & Applied [email protected]
IP4
Granular Experiments of Discrete Nonlinear Sys-tems
Ordered and disordered arrangements of granular particlesare governed by highly nonlinear contact interactions thatconfer to the particles assembly unique dynamic properties.We study how stress waves propagate in lattices composedof particles in close contact, and exploit this understand-ing to create novel materials and devices at different scales(for example, for application in energy conversion, vibra-tion absorption and acoustic rectification). We control theconstitutive behavior of these new materials selecting theparticles geometry, their arrangement and materials prop-erties. We assemble and test experimental systems and in-form our experiments with discrete numerical simulations.
Chiara DaraioETH Zurich
PD13 Abstracts 57
IP5
Tug-of-war and Infinity Laplacian with NeumannBoundary Conditions
We study a version of the stochastic ”tug-of-war” game,played on graphs and smooth domains, with an emptyset of terminal states. We prove that, when the run-ning payoff function is shifted by an appropriate con-stant, the values of the game after n steps converge. Us-ing this we prove the existence of solutions to the in-finity Laplace equation with vanishing Neumann bound-ary condition. In earlier work with Schramm, Sheffieldand Wilson (http://arxiv.org/abs/math/0605002, JAMS2009), we related a tug of war game to the infinity Lapla-cian equation with Dirichlet boundary conditions- I willsurvey that work as well as the version for the p-Laplacianin http://arxiv.org/abs/math/0607761 - Duke 2009. (Talkbased on joint work with Tonci Antunovic, Scott Sheffield,Stephanie Somersille http://arxiv.org/abs/1109.4918 ,Comm. PDE 2012)
Yuval PeresMicrosoft [email protected]
IP6
Models for Neural Networks; Analysis, Simulationsand Qualitative Behavior
Neurons exchange information via discharges propagatedby membrane potential which trigger firing of the manyconnected neurons. How to describe large networks of suchneurons? How can such a network generate a collectiveactivity? Such questions can be tackled using nonlinearpartial-integro-differential equations which are classicallyused to describe neuronal networks. Among them, theWilson-Cowan equations are the best known and describeglobally brain spiking rates. Another classical model is theintegrate-and-fire equation based on Fokker-Planck equa-tions. The spike times distribution, which encodes moredirectly the neuronal information, can also be describeddirectly thanks to structured population. We will compareand analyze these models. A striking observation is thatsolutions to the I&F can blow-up in finite time, a form ofsynchronization that can be regularized with a refractorystage. We can also show that for small or large connectiv-ities the ’elapsed time model’ leads to desynchronization.For intermediate regimes, sustained periodic activity oc-curs compatible with observations. A common tool is theuse of the relative entropy method.
Benoit PerthameUniversite Pierre et Marie CurieLaboratoire Jacques-Louis Lions [email protected]
IP7
A PDE Approach to Computing Viscosity Solu-tions of the Monge-Kantorovich Problem
After a quick overview of the optimal transport for theEuclidean distance problem and available numerical meth-ods, I will present a new technique to deal with the stateconstraint that binds the transport when source and tar-get have compact support. It takes the form of non-linear
boundary conditions which can be combined to a Monge-Ampre equation to solve the optimal transport problem.The wide-stencil discretization technique and fast Newtonsolver proposed by Oberman and Froese is extended to thisframework and allows to compute weak viscosity solutionof the optimal transport problem. Numerical solutions willbe presented to illustrate strengths and weaknesses of themethod.
Jean-David BenamouINRIA Rocquencourt, [email protected]
IP8
Modelling Collective Cell Motion in Biology
We will consider three different examples of collective cellmovement which require different modelling approaches:movement of cells in epithelial sheets, with application torosette formation in the mouse epidermis and monoclonalconversion in intestinal crypts; cranial neural crest cell mi-gration which requires a hybrid discrete cell-based chemo-taxis model; acid-mediated cancer cell invasion, modelledvia a coupled system of non-linear partial differential equa-tions. We show that in many cases, all these models canbe expressed in the framework of nonlinear diffusion equa-tions. We show how these models can be used to under-stand a range of biological phenomena.
Philip K. MainiCentre for Mathematical BiologyUniversity of [email protected]
CP1
A Comparison Analysis of q-Homotopy Analy-sis Method (q-HAM) and Variational IterationMethod (VIM) to Fingero Imbibition Phenomenain Double Phase Flow through Porous Media
In this paper, we consider Variational Iteration Method(VIM) and q-Homotopy Analysis Method (q-HAM) tosolve the partial differential equation resulted from FingeroImbibition phenomena in double phase flow through porousmedia. We further compare the results obtained here withthe solution obtained in [?] using Adomian DecompositionMethod. Numerical results are obtained, using Mathemat-ica 9, to show the effectiveness of these methods on ourchoice of problem especially for suitable values of hand n.
Olaniyi S. IyiolaKing Fahd University of Petroleum and [email protected]
CP1
Weak Solutions to Lubrication Systems Describingthe Evolution of Bilayer Thin Films
We prove existence of global non-negative weak solutionsfor coupled one-dimensional lubrication systems that de-scribe the evolution of nanoscopic bilayer thin polymerfilms. We consider Navier-slip and no-slip conditions atboth liquid-liquid and liquid-solid interfaces. Additionally,we show existence of positive smooth solutions when at-tractive van der Waals and repulsive Born intermolecularinteractions are taken into account.
Sebastian JachalskiWeierstrass Institute
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Georgy KitavtsevMax Planck Institute for Mathematics in the [email protected]
Roman TaranetsInstitute of Applied Mathematics and Mechanics NASUkrainetaranets [email protected]
CP1
Stability Analysis of Thin Film Problems withNon-Constant Base States
We address the linear stability of growing rims that appearin thin solid and liquid films as they retract from a solidsubstrate. Despite the different transport mechanisms, themathematical models in both cases lead to mass conservingfree boundary problems for degenerate and non-degeneratethin film equations of the type
ht +∇ · (hn∇Δh) = 0,
with different mobilities hn. The base state is time de-pendent and does not have a simple traveling wave or self-similar form and thus prevents a straight forward linearstability analysis. However, large rims evolve on a slowertime scale than the perturbations. We exploit this timescale separation to derive asymptotic approximations forthe base states and obtain solutions of the linear stabilityproblem. Our results enable to track the evolution of thedominant wavelength and yield criteria for the validity ofthe so-called “frozen modes analysis’ often used for thesetypes of problems.
Dziwnik MarionTU BerlinInstitut fur [email protected]
Maciek KorzecTU [email protected]
Andreas MuenchOCIAM Mathematical InstituteUniversity of [email protected]
Barbara WagnerWeierstrass Institute, BerlinTU [email protected]
CP1
A New Mixed Formulation For a Sharp InterfaceModel of Stokes Flow and Moving Contact Lines
Two phase fluid flows on substrates (i.e. wetting phenom-ena) are important in many industrial processes, such asmicro-fluidics and coating flows. These flows include addi-tional physical effects that occur near moving (three-phase)contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension(see Falk, Walker in the context of Hele-Shaw flow) thatinteracts with a rigid substrate. The model is derived byan Onsager type principle using shape differential calculus(at the sharp-interface, front-tracking level) and allows for
moving contact lines and contact angle hysteresis througha variational inequality. We prove the well-posedness ofthe time semi-discrete and fully discrete (finite element)model and discuss error estimates. Simulation movies willbe presented to illustrate the method. We conclude withsome discussion of a 3-D version of the problem as well asfuture work on optimal control of these types of flows.
Shawn WalkerLouisiana State UniversityDepartment of Mathematics and [email protected]
CP2
Two-Point Riemann Problem for InhomogeneousNonconvex Conservation Laws: Geometric Con-struction of Solutions
We consider the following conservation law: ∂tu(x, t) +∂xf(u(x, t)) = g(u) The following boundary conditions arespecified: u(0 − 0, t) = u0− and u(X + 0, t) = uX+. Inaddition, we specify the initial condition u(x, 0) = ux0 forx ∈ (0, X) Method of characteristics is used to show theevolution of the initial profile and the discontinuities atthe boundaries as well as appearance and in some casesdisappearance of internal discontinuities. For illustrationpurposes the function f(u) will be assumed to have 3 crit-ical points.
Dmitry A. AltshullerDassault [email protected]
CP2
A Nonlocal Hyperbolic Pde for Chaotic Shocks
We propose a new model equation that describes chaoticshock waves. The equation is a simple modification of theBurgers equation that includes non-locality via the pres-ence of the shock-state value of the solution in the equa-tion itself. The model predicts steady-state solutions, theirinstability through Hopf bifurcation, and a sequence ofperiod-doubling bifurcations leading to chaos. The dynam-ics of the solutions is qualitatively identical to those in theone-dimensional reactive Euler equations. We present acomplete linear stability theory as well as nonlinear nu-merical simulations to characterize the observed chaoticattractor.
Aslan R. KasimovDivision of Mathematical & Computer Sciences andEngineeringKing Abdullah University of Science and [email protected]
Luiz [email protected]
Rodolfo R. RosalesMassachusetts Inst of TechDepartment of [email protected]
CP2
A Riemann Problem at a Junction of Open Canals
We study a Riemann problem at a junction of a star-like
PD13 Abstracts 59
network of open canals. The flow in the network is givenby 1D Saint-Venant equations in each canal and specialconditions at the junction. We consider the case wheretwo of the canals are identical. Firstly, we show that thelinearised problem has always a unique solution. Secondly,we show that under certain condition, there is a uniquesolution to the nonlinear Riemann problem.
Mouhamadou S GoudiabyLANI, UFR SAT Universite Gaston BergerBP 234, Saint-Louis, [email protected]
Gunilla KreissDivision of Scientific ComputingUppsala [email protected]
CP2
Stability of Viscous Strong and Weak DetonationWaves for Majda’s Model
We give an overview of a program based on a combina-tion of analytical and numerical Evans-function techniquesfor establishing the nonlinear stability of strong and weakdetonation-wave solutions of Majda’s simplified, “qualita-tive’ model for reacting mixtures of gases. One noteworthyaspect of the analysis is the treatment of weak viscous det-onation waves. Due to to their nature, the stability of theseundercompressive waves has received scant attention in theliterature.
Gregory LyngDepartment of MathematicsUniversity of [email protected]
CP2
Transitional Wave-Dynamics forHyperbolic Relaxation Systems
We consider the transition between the wave dynamics ofthe homogeneous relaxation system and that of the localequilibrium approximation for a linearized hyperbolic re-laxation system. As a specific example, we look at a Euler-type two-phase flow model with relaxation towards phaseequilibrium. We observe that the stability of the transi-tional waves is equivalent to the sub-characteristic condi-tion and identify the existence of a critical region of wavenumbers where the sonic waves disappear.
Susanne Solem, Peder AursandNorwegian University of Science and [email protected], [email protected]
Tore FlattenSINTEF Materials and [email protected]
CP2
Glancing Weak Mach Reflection
We study the glancing limit of weak shock reflection, inwhich the wedge angle tends to zero with the Mach num-ber fixed. Lighthill showed that, according to linearizedtheory, the reflected shock strength approaches zero at thetriple point in reflections of this type. To understand thenonlinear structure of the solution near the triple point in
nearly glancing reflections, then, it is necessary to under-stand how the reflected shock diffracts nonlinearly into theMach shock as its strength approaches zero. Towards thisend, we formulate a half-space initial boundary value prob-lem for the unsteady transonic small disturbance equationsthat describes nearly glancing Mach reflection. We solvethis IBVP numerically using high-resolution methods, andwe find in the solutions a complex reflection pattern thatclosely resembles Guderley Mach reflection. This is jointwork with John Hunter.
Allen TesdallDepartment of MathematicsCity University of New York, College of Staten [email protected]
CP3
Reconstruction of Nonlinear Water Waves by Gen-eralized SfS Method
We introduce a generalized nonlinear PDE for the reflec-tivity function which occurs in optical reconstruction ofdynamical surfaces. We use a Lambertian reflectance SfSprocedure for the reconstruction of water waves. We com-pared the results of our numerical processing of data withexperiments in the wave lab at ERAU, for different typesof surface waves: linear, nonlinear, rogue, vortex induced,etc. We double check our SfS results with laser goniometryand capacitive level sensors.
Andrei LuduEmbry-Riddle Aeronautical UniversityMathematics [email protected]
Nikhila Chaudhari, Muhammad Abdul Aziz, IlterisDemirkiranEmbry-Riddle Aeronautical UniversityDept. Electrical, Computer, Software, & [email protected], [email protected],[email protected]
CP3
Existence and Symmetry of Ground States to theBoussinesq Abcd Systems
We consider a four-parameter family of Boussinesq systemsderived by Bona-Chen-Saut. We establish the existence ofthe ground states which are solitary waves minimizing theaction functional of the systems. We further show that inthe presence of large surface tension the ground states areeven up to translation.
Ming Chen, Shiting Bao, Qing LiuUniversity of [email protected], [email protected], [email protected]
CP3
Sharp Thresholds of Global Existence and Blow-Upfor a Class of Nonlocal Wave Equations
We first present the local and global existence and finite-time blow-up results for the nonlocal nonlinear wave equa-tions utt − Kuxx + Mutt = g(u)xx, where K and Mare two pseudo-differential operators. We then establishthresholds for global existence versus blow-up in the caseof power-type nonlinearities. We use the potential well
60 PD13 Abstracts
method based on the concepts of invariant sets suggestedby Payne and Sattinger. The results cover those givenfor the so-called double-dispersion equation and the tra-ditional Boussinesq-type equations, as special cases. Thiswork has been supported by the Scientific and Technolog-ical Research Council of Turkey (TUBITAK) under theproject TBAG-110R002.
Saadet ErbayOzyegin [email protected]
Albert ErkipSabanci [email protected]
Husnu A. ErbayOzyegin [email protected]
CP3
Stability and Instability of Solitary Waves for aClass of Nonlocal Nonlinear Equations
We investigate the existence and stability/instability prop-erties of solitary waves for the general class of non-local nonlinear wave equations utt − Kuxx + Mutt =± (|u|p−1u
)xx
, (p > 1), where K and M are two pseudo-differential operators. The so-called double-dispersionequation and the Boussinesq-type equations are two well-known members of this class. The relative dominance ofthe two operators M and K plays an important role inour investigations. This work has been supported by theScientific and Technological Research Council of Turkey(TUBITAK) under the project TBAG-110R002.
Husnu A. ErbayOzyegin [email protected]
Albert ErkipSabanci [email protected]
Saadet ErbayOzyegin [email protected]
CP3
On the Generalized KdV Equation
In this talk we consider the generalized Korteweg-de Vries(gKdV) equation
∂tu+ ∂3xu+ μ∂x(u
k+1) = 0,
where k ≥ 4 is an integer number and μ = ±1. We willreview the well-posedness theory and discuss some openproblems.
Luiz G. FarahUniversidade Federal de Minas Gerais (UFMG) - [email protected]
CP3
Orbital Stability of Solitary Waves of ModerateAmplitude in Shallow Water
We study the orbital stability of solitary traveling wave
solutions of the following equation for surface water wavesof moderate amplitude in the shallow water regime:
ut+ux+6uux−6u2ux+12u3ux+uxxx−uxxt+14uuxxx+28uxuxx = 0.
Our approach is based on a method proposed by Grillakis,Shatah and Strauss in 1987, and relies on a reformulationof the evolution equation in Hamiltonian form. We deducestability of solitary waves by proving the convexity of ascalar function, which is based on two nonlinear functionalsthat are preserved under the flow.
Nilay Duruk MutlubasSabanci [email protected]
Anna GeyerUniversity of [email protected]
CP3
Eulerian Computation Of Complex Short WaveForms Propagating Over Long Distances
We present a new Eulerian method for computation ofshort waveforms over long distances using Nonlinear Soli-tary Waves that are solutions to the Hamilton-Jacobi equa-tion. This method overcomes limitations of conventionaldiscretization schemes, whose accuracy depends on gridresolution, making them too costly for real world problems.This cost can be greatly reduced by carrying the waveformsusing thin pulses, which span over 4 grid cells and implic-itly specifying the waveforms through initial conditions.
Subhashini [email protected]
John SteinhoffUniversity of Tennessee Space Inst.Goethert [email protected]
CP4
On Inviscid Limits for the Stochastic Navier-StokesEquations and Related Systems.
One of the original motivations for the development ofstochastic partial differential equations traces it’s originsto the study of turbulence. In particular, invariant mea-sures provide a canonical mathematical object connectingthe basic equations of fluid dynamics to the statistical prop-erties of turbulent flows. In this talk we discuss some recentresults concerning inviscid limits in this class of measuresfor the stochastic Navier-Stokes equations and other re-lated systems arising in geophysical and numerical settings.This is joint work with Peter Constantin, Vladimir Sverakand Vlad Vicol.
Nathan Glatt-HoltzVirginia [email protected]
CP4
Rethinking Computation of Incompressible FluidFlow More Than a Mathematical Trick
The Navier-Stokes equation is composite, orthogonally de-composable into a (dependent) pressure and a fundamental
PD13 Abstracts 61
pressure-free governing equation for incompressible flow.In differential form, these involve integrals over a Greensfunction for the Laplacian operator, but its not immedi-ately obvious how to solve them numerically. However,the equivalent variational forms are amenable to efficientnumeric computation with a simple finite element methodusing discrete divergence-free velocity bases found as thecurl of modified Hermite elements.
Jonas T. [email protected]
CP4
Compressible Viscous Navier-Stokes Flows Grazinga Non-Convex Corner
I will talk about existence and regularity of solution for thecompressible viscous Navier-Stokes equations on a polygonhaving a grazing non-convex corner. I will also talk abouta jump discontinuity of the density function across a curveinside the domain and the corresponding jumps in deriva-tives of the velocity. The solution comes from a well-posedboundary value problem on a non-convex polygonal do-main. By the decay formula of the jump it is seen thatdensity jumps can happen in a high speed flow occuring ina very viscous compressible fluid.
Jae Ryong Kweon
POSTECH(Pohang University of Science and Technology)[email protected]
CP4
Entropy-Stable Schemes for the Initial BoundaryValue Euler Equations
We consider numerical schemes for the nonlinear Eulerequations of gas dynamics in one space dimension. Fi-nite difference numerical schemes, which satisfy an entropystability condition, have been derived and include far-fieldand wall boundary conditions. We have also been deriveda stable numerical scheme for connecting two domains viaan interface. Furthermore, we present numerical compu-tations for second and fourth order accurate schemes todemonstrate robustness of the proposed schemes.
Hatice OzcanSchool of Mathematics, University of Edinburgh,JCMBKing’s Buildings, Mayfield Road, Edinburgh, EH9 [email protected]
Magnus SvardUniversity of Bergen, Department of [email protected]
CP4
Primitive Equations with Continuous Initial Data
Weaddressthewell-posednessoftheprimitiveequationsoftheoceanwithcontinuousinitialdata. We show that the splitting of the initial data intoa regular finite energy part and a small bounded part ispreserved by the equations thus leading to existence anduniqueness of solutions.
Walter RusinUniversity of Southern [email protected]
Walter RusinOklahoma State [email protected]
Igor Kukavica, Yuan Pei, Mohammed ZianeUniversity of Southern [email protected], [email protected], [email protected]
CP4
A Unilateral Open Boundary Condition for theNavier-Stokes Equations
One of the main issues in the simulations of blood flow inarteries is a proper setting of the outflow boundary condi-tion at artificial boundaries. Under some physical assump-tions, we propose a new approach to set the outflow bound-ary condition by using a variational inequality technique.Consequently, we solve the time-dependent Navier-Stokesequation with a boundary-penalty term. We report somemathematical and numerical results on this new problem,and compare it with other methods.
Norikazu SaitoUniversity of TokyoGraduate School of Mathematical [email protected]
Guanyu ZhouGraduate School of Mathematical Sciences,The University of [email protected]
Yoshiki SugitaniThe University of TokyoGraduate School of Mathematical [email protected]
CP4
Existence of Classical Sonic-Supersonic Solutionsfor the Steady Euler Equations
Given a smooth curve as a sonic line in the plane, we con-struct a classical sonic-supersonic solution on one side ofthe curve for the steady isentropic compressible Euler sys-tem in two space dimensions. The solution is obtained byemploying an innovate coordinate system. The streamlinesof the equations are analyzed to obtain a physical domainof existence.
Tianyou ZhangThe Pennsylvania State Universityuniversity [email protected]
Yuxi ZhengPenn State [email protected]
CP5
Multiple Steady State Solutions of Some Reaction-Diffusion Systems
We consider a class of reaction-diffusion systems leading tostrongly indefinite energy functionals, with nonlinearitieswhich combine concave and convex terms. By establishingtwo new critical point theorems for strongly indefinite evenfunctionals, we obtain the existence of infinitely many largeand small energy steady state solutions. Our critical point
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theorems generalize the fountain theorems of Barstch andWillem to strongly indefinite functionals.
Cyril Joel BatkamUniversity of [email protected]
CP5
Phase Transitions with Mid-Range Interactions: aNon-Local Stefan Model
We study a non-local version of the one-phase Stefan prob-lem which takes into account mid-range interactions, whichlead to new phenomena which are not present in the usuallocal version of the one-phase Stefan model: creation ofmushy regions, existence of waiting times during which theliquid region does not move, and possibility of melting nu-cleation. If the kernel is suitably rescaled, the correspond-ing solutions converge to the solution of the local one-phaseStefan problem.
Cristina BrandleU. Carlos III [email protected]
Emmanuel ChasseigneU. Franois Rabelais (Tours)[email protected]
Fernando QuirosU. Autonoma [email protected]
CP5
Stability and Convergence Analysis for NonlocalDiffusion and Nonlocal Wave Equations
We consider the time-dependent nonlocal diffusion andnonlocal wave equations, formulated in the nonlocal peri-dynamics setting. Initial and boundary data are given.For nonlocal diffusion equation, the time derivatives areapproximated using Forward Euler, Backward Euler andCrank–Nicholson schemes. For nonlocal wave equation, weget the dispersion relations and use the Newmark methodto discretize the equation. And we use finite elementmethod to discretize the spatial domain. For both equa-tions, the standard timestep stability conditions must bereformulated, in light of the peridynamics formulation. Wehave got the the stability conditions and the convergencetheorems. Computational implications verify our results.
Qingguang GuanDepartment of Scientific ComputingFlorida State [email protected]
Max GunzburgerFlorida State UniversitySchool for Computational [email protected]
CP5
Localized Perturbations of the Complex Ginzburg-Landau Equation: Rcovering Fredholm PopertiesVia Kondratiev Spaces
We consider perturbations to the complex Ginzburg-Landau equation in dimension 3: At = (1+iα)ΔA+A−(1+
iγ)A|A|2 + iεg(x)A , where g(x) is a localized real valuedfunction and ε is small. We are interested in finding solu-tions close to patterns of the form A = e−iγtA. At ε = 0 thelinearization about the equilibrium A∗ = 1 is not invert-ible in the usual translation invariant spaces. We show thatwhen considered as an operator between Kondratiev spacesthe linearization is a Fredholm operator. These spaces con-sist of functions with algebraic localization that increaseswith each derivative. We use this result to construct so-lutions close to the equilibrium via the Implicit FunctionTheorem and derive asymptotics for wavenumbers in thefar field. This is joint work with Arnd Scheel.
Gabriela JaramilloUniversity of [email protected]
CP5
Global Attractors of the Hyperbolic Relaxationof Reaction-Diffusion Equations with DynamicBoundary Conditions
Under consideration is the hyperbolic relaxation of thesemilinear reaction-diffusion equation,
εutt + ut −Δu+ f(u) = 0,
ε ∈ [0, 1], with the prescribed dynamic boundary condition,
∂nu+ u+ ut = 0.
For all singular and nonsingular values of the perturbationparameter, we obtain global attractors with optimal regu-larity. After fitting both problems into a common frame-work, a proof of the upper-semicontinuity of the family ofglobal attractors is given. The result is motivated by theseminal work of J. Hale and G. Raugel in Upper semiconti-nuity of the attractor for a singularly perturbed hyperbolicequation, J. Differential Equations 73 (1988).
Joseph L. ShombergProvidence [email protected]
Ciprian GalFlorida International [email protected]
CP5
On λ-Symmetry Classification and ConservationsLaws of Differential Equations
We study λ-symmetry classification of nonlinear differen-tial equations based on the fin equation. The λ-symmetryapproach allows us the finding of integrating factors, theconservation forms as first integrals, similarity solutionsand reducing degrees of differential equation, which is anew approach in the theory of Lie groups. We presentalso the theory for computing the integrating factors fromλ-symmetries for a second order ordinary differential equa-tions. Then the symmetry classifications are constructedfor the different forms of heat transfer coefficient and ther-mal conductivity functions of the nonlinear fin equations.In addition, as a different approach the Jacobi last mul-tiplier method is considered to determine the new formsof λ-symmetries and corresponding conservations laws andinvariant solutions of the nonlinear fin equations and theresults obtained from two different methods are comparedin the study.
Teoman zer, zlem Orhan, Gulden Gun
PD13 Abstracts 63
Istanbul Technical [email protected], [email protected], [email protected]
CP6
Mathematical Analysis on Chevron Structures inLiquid Crystal Films
In this presentation, I will talk about the mathematicalanalysis of a model for the chevron structure arising fromthe cooling from the Smectic-A liquid crystal to the chiralSmectic-C phase in a surface-stabilized cell with certainboundary conditions under a given electric field. This phe-nomenon causes severe defects in liquid crystal display de-vices and has attracted interest from both theoretical andpractical point of view. In the static analysis, the stabilityof the chevron structure is established through a sequenceof minimization problems converging to a reduced energyfunctional based on Chen-Lubensky model. This study es-tablishes the existence of minimizers and provides analysisof the minimal energy configuration in the limiting case.
Lei Z. ChengIndiana University [email protected]
CP6
On the Existence of Absolute Weak Minimizers ofEnergy Functionals Associated with the BoundaryValue Problem of Nonlinear Elasticity
In this communication, I will present a new necessary andsufficient condition for the existence of absolute weak min-imizers that applies to scale-invariant energy functionalsincluding those appearing in the pioneering work of 1982by J. Ball on the pure displacement boundary value prob-lem of nonlinear hyper-elastiicity. This result also bringsnew ideas to the regularity question associated with theboundary value problem of nonlinear hyper-elasticity thatwould replace the delicate phase plane analysis.
Salim M. HaidarGRAND VALLEY STATE [email protected]
CP6
Multicomponent Polymer Flooding in Two Dimen-sional Oil Reservoir Simulation
We propose a high resolution finite volume scheme for a(m+1)(m+1) system of nonstrictly hy perbolic conserva-tion laws which models multicomponent polymer floodingin enhanced oil-recovery process in two dimensions. In thepresence of gravity the flux functions need not be mono-tone and hence the exact Riemann problem is complicatedand computationally expensive. To overcome this diffi-culty, we use the idea of discontinuous flux to reduce thecoupled system into uncoupled system of scalar conserva-tion laws with discontinuous coefficients. High order ac-curate scheme is constructed by introducing slope limiterin space variable and a strong stability preserving Runge-Kutta scheme in the time variable. The performance ofthe numerical scheme is presented in various situations bychoosing a heavily heterogeneous hard rock type medium.Also the significance of dissolving multiple polymers inaqueous phase is presented
Sudarshan Kumar Kenettinkara, Praveen c, G.D.Veerappa GowdaTIFR Centre For Applicable Mathematics,India
[email protected],[email protected], [email protected]
CP6
A Longwave Model for Strongly AnisotropicGrowth of a Crystal Step
A model of morphological evolution of a single growingcrystal step is presented. Via a multi-scale expansion,we derived a long-wave, strongly nonlinear, and stronglyanisotropic evolution PDE for the step profile, performedthe linear stability analysis and computed the nonlineardynamics. Linear stability depends on whether the stiff-ness is minimum or maximum in the direction of the stepgrowth. It also depends nontrivially on the combinationof the anisotropy strength parameter and the atomic fluxfrom the terrace to the step. Computations show forma-tion and coarsening of a hill-and-valley structure superim-posed onto a large-wavelength profile, which independentlycoarsens. Coarsening laws for the hill-and-valley structureare computed for both orientations of a maximum step stiff-ness, the increasing anisotropy strength, and the varyingatomic flux.
Mikhail V. KhennerDepartment of MathematicsWestern Kentucky [email protected]
CP6
Finite-Volume Numerical Approximations of Con-servation Law with Fading Memory
We study the longitudinal motion of materials where thedestabilizing influence of nonlinear elastic response com-petes with the damping effect of the fading memory. Theresulting mathematical model is a system of conservationlaws involving a time convolution integral. We construct,implement, and analyze a new numerical method based onthe finite volume method. The implementation relies onnumerical methods for advection equations with (discrete)delay. Finally we apply the complete algorithm to the elas-tic memory problem to make quantitatively predictions forthe behaviour of the material.
Paramjeet SinghPanjab University, Chandigarh, [email protected]
CP6
Acoustic Resonance of a Vapor Bounded by ItsOwn Liquid Phase
The pressure and temperature perturbations due to acous-tic waves in an initially quiescent vapor in equilibrium withits liquid phase induce the evaporation and condensation atthe vapor-liquid interface. The phenomenon can be math-ematically formulated by a set of fluid-dynamics equationsfor vapor and liquid and boundary conditions at the in-terface. The boundary-value problem is solved and somephysical features of solution for acoustic resonance involv-ing the evaporation and condensation at the interface arediscussed.
Takeru YanoDepartment of Mechanical Engineering,Osaka University
64 PD13 Abstracts
CP7
The Robin Eigenvalue Problem for the p(x)-Laplacian As p → ∞We study the asymptotic behavior, as p → ∞, of thefirst eigenvalues and the corresponding eigenfunctions forthe p(x)-Laplacian with Robin boundary conditions in anopen, bounded domain Ω ⊂ R
N with sufficiently smoothboundary. We prove that the positive first eigenfunctionsconverge uniformly in Ω to a viscosity solution of a prob-lem involving the ∞-Laplacian with appropriate bound-ary conditions. Joint work with F. Abdullayev (WorcesterPolytechnic Institute).
Marian BoceaLoyola University [email protected]
CP7
Well-Posedness of a Mathematical Model for TraceGas Sensors
Trace gas sensors have great potential in many areas suchas monitoring atmospheric pollutants, leak detection, anddisease diagnosis through breath tests. We present a math-ematical model of such a sensor with a modulated lasersource leading to a complex system of elliptic partial differ-ential equations. Here we present a proof of well-posednessfor the resulting system of equations via the Lax-Milgramtheorem as well as error estimates of the finite elementmethod solution.
Brian W. Brennan, Robert C. KirbyBaylor Universityb [email protected], Robert [email protected]
CP7
A Unique Solution to a Quasi-Linear Elliptic Equa-tion
We prove the existence of a unique classical solution u tothe equation −∇ · (a(u)∇u) + v · ∇u = f with boundarycondition ∇u · n = g, where the value of the solution u(x)is known at a point x0 in the domain. The key to provingthe uniqueness of the solution is an a priori estimate for uin a Sobolev space norm.
Diane DennyTexas A&M University - Corpus [email protected]
CP7
Eigenvalues and Eigenfunctions of the LaplacianVia Inverse Iteration with Shift
We present an iterative method, inspired by the inverseiteration with shift technique of finite linear algebra, de-signed to find the eigenvalues and eigenfunctions of theLaplacian with homogeneous Dirichlet boundary conditionfor arbitrary bounded domains Ω ⊂ R
N . Uniform con-vergence away from nodal surfaces is also obtained. Themethod can also be used in order to produce the spectraldecomposition of any given function u ∈ L2(Ω).
Eder M. Martins
UFOP - UNIVERSIDADE FEDERAL DE [email protected]
Grey Ercole, Rodney Biezuner, Breno GiacchiniUFMG - UNIVERSIDADE FEDERAL DE [email protected], [email protected],[email protected]
CP7
Global Regularity for a Class of Quasi-Linear Non-local Elliptic Equations
We consider the solvability of quasi-linear elliptic equationswith either nonlocal Neumann, Robin, or Wentzell bound-ary conditions, defined on bounded non-smooth domains.We develop the corresponding regularity theory for weaksolutions, and show that such weak solutions are globallyHolder continuous. Several motivations, applications, andother properties are also addressed.
Alejandro Velez-SantiagoDepartment of MathematicsUniversity of California, [email protected]
CP8
Fractional Differential Equations with Impulses
In this talk we discuss fractional order impulsive differen-tial equations. We establish existence and uniqueness ofthe solution using fixed point techniques. Moreover, wealso apply our analytical result to fractional order model.At the end we prove the stability and perform some numer-ical simulation to give a pictorial view of the theoreticalfindings.
Syed AbbasInidan Institute of technology Mandi, [email protected]
CP8
Stability of Solutions of Parabolic and HyperbolicSystems of Partial Differential Equations
The lecture is devoted to the finding of criteria of solutionsstability of parabolic and hyperbolic systems of differen-tial equations. We consider the nonlinear parabolic sys-tems with Riemann-Liouville fractional derivatives. Thetechnique of stability criteria finding consists of integraltransformations of the considered equations into the sys-tems of ordinary differential equations and investigationthe connection between stability of the systems of ordinarydifferential equations and systems of parabolic equations.For the investigation of solutions stability of systems ofordinary differential equations with time-dependend coeffi-cients we use methods given in [1]. [1] Boykov I.V. Stabilityof solutions of differential equations. Penza: Penza StateUniversity. 2008. 244 p.
Ilya V. Boykov, Vladimir RyazantsevPenza State [email protected], [email protected]
CP8
Exact Solutions of Nonlinear Fractional Partial Dif-
PD13 Abstracts 65
ferential Equation
Oldman and Spanier first considered the fractional diferen-tial equations arising in diffusion problems. The fractionaldiferential equations have been investigated by many au-thors. In recent years, some effective methods for fractionalcalculus were appeared in open literature, such as the frac-tional sub-equation method and the first integral method.The fractional complex transform is the simplest approach,it is to convert the fractional diferential equations into ordi-nary diferential equations, making the solution procedureextremely simple. Recently, the fractional complex trans-form has been suggested to convert fractional order differ-ential equations with modified Riemann-Liouville deriva-tives into integer order diferential equations, and the re-duced equations can be solved by symbolic computation.The present paper investigates for the applicability andefficiency of the exp-function method on some fractionalnonlinear diferential equations.
Adem CevikelYildiz Technical [email protected]
Ozkan GunerDumlupinar [email protected]
Ahmet BekirEskisehir Osmangazi [email protected]
Mutlu AkarYildiz Technical [email protected]
CP8
Multiscaling Modelling in Evolutionary Dynamics
We study a class of processes that are akin to the Wright-Fisher model, with transition probabilities weighted interms of the frequency-dependent fitness of the popula-tion types. By considering an approximate weak formula-tion of the discrete problem, we are derive a correspondingcontinuous weak formulation for the probability density.Therefore, we obtain a family of PDEs for the evolution ofthe probability density, and which will be an approxima-tion of the discrete process in the joint large population,small time-steps and weak selection limit. The equationsin this family can be purely diffusive, purely hyperbolicor of convection-diffusion type, with frequency dependentconvection. The diffusive equations are of the degener-ate type If the fitness functions are sufficiently regular, wecan recast the weak formulation in a stronger formulation,without any boundary conditions, but supplemented by anumber of conservation laws.
Max O. SouzaDepartamento de Matematica AplicadaUniversidade Federal [email protected]
Fabio ChalubUniversidade Nova de [email protected]
CP8
A Game Theoretic Approach to Non-Relativistic
Quantum Mechanics
Research into quantum foundations has led to the develop-ment of a model of NRQM based upon process theory. Pro-cesses are modeled by means of a two player, co-operative,combinatorial, forcing game, generating a discrete collec-tion of elements embedding into a space-like hyspersur-face in R
4, evolving discretely forward in time. Each el-ement is associated with a locally generated sinc wavelet,the NRQM wave function being the interpolation over thehypersurface. This results in a model of NRQM which islocal and non-contextual at the primitive level, yet non-local and contextual at the process level. The structure ofthe game will be described.
William H. SulisMcMaster UniversityDepartments of Psychiatry and [email protected]
CP9
Numerical Methods and Multi-Scale Analysis forthe Dirac Equation in Nonrelativistic Limit Regime
We compare numerically temporal/ spatial resolution ofvarious numerical methods for solving the Dirac equationin nonrelativistic limit regime, involving a small param-eter 0 < ε ≤ 1 which is inversely proportional to thespeed of light. In this regime, the solution is highly os-cillating in time, which significantly increase the com-putational burden. Using the conventional finite differ-ence methods, the time step τ requires a ε−scalability:τ = O(ε3). Then we propose some new methods, TimeSplitting Spectral(TPPS) and Exponential Wave Integrat-ing(EWI). The new methods are unconditionally stableand their ε−scalability are improved to τ = O(ε2). At lastwe provide the multi-scale method for the Dirac equationto obtain a better numerical solution.
Xiaowei JiaNational University of [email protected]
CP9
On the Method of Numerical Integration for Fric-tion Boundary Conditions
Friction boundary conditions are formulated by variationalinequalities of second kind, where the nonlinearity is rep-resented by an L1 norm on the boundary. We show thatsome numerical integration approximation for the L1 normoffers nice properties such as a discrete version of comple-menting conditions. Applications to the Poisson, elasticity,and fluid equations are considered.
Takahito KashiwabaraGraduate School of Mathematical SciencesThe University of [email protected]
CP9
Finite Element Analysis of the Stationary Power-Law Stokes Equations Driven by Friction BoundaryConditions
We are concern with the finite element approximation forthe stationary power law Stokes equations driven by slipboundary condition of “friction type”. It is shown that byapplying a variant of Babuska-Brezzi’s theory for mixed
66 PD13 Abstracts
problems, convergence of the finite element approximationformulated is achieved with classical assumptions on theregularity of the weak solution. Solution algorithm for themixed variational problem is presented and analyzed indetails. Finally, numerical simulations that validate thetheoretical findings are exhibited.
Mbehou MohamedDepartment of Mathematics and Applied MathsUniversity of Pretoria, South [email protected]
Djoko Kamdem JulesDepartment of Mathematics and Applied MathematicsUniversity of Pretoria, South [email protected]
CP9
Non-Linear Controller of Combined RadiativeConduction Systems
This correspondence studies the problem of finite dimen-sional non-linear control for a class of systems described bynonlinear hyperbolic-parabolic coupled partial differentialequations (PDEs). Initially, Galerkins method is applied tothe PDE system to derive a nonlinear ordinary differentialequation (ODE) system that accurately describes the dy-namics of the dominant (slow) modes of the PDE system.After, we introduce a useful non linear controller to assurestabilization under convex sufficient conditions. Numericalexamples show high performances.
Ghattassi Mohamed, Ghattassi MohamedUniversite de Lorraine, Institut Elie Cartan de Lorraine,[email protected],[email protected]
CP9
A Second-Order Time Discretization Scheme for aSystem of Nonlinear Schrodinger Equations
We propose a new time discretization scheme for a sys-tem of nonlinear Schrodinger equations which is a model ofthe interaction of a non-relativistic particles with differentmasses. Our scheme is composed of two (complex-valued)linear systems at each time step, and the solution is shownto converge at a second order rate. We report numericalexample to confirm the theoretical results. Our idea canbe applied to a large class of nonlinear PDEs.
Takiko SasakiGraduate School of Mathematical SciencesThe University of [email protected]
Norikazu SaitoUniversity of TokyoGraduate School of Mathematical [email protected]
CP9
Fictitious Domain Method with the L2-Penalty andApplication to the Finite Element and Finite Vol-ume Methods
The fictitious domain approach is useful to compute nu-merical solutions of PDEs defined in complex domains andtime-dependent moving domains. In this paper, we study
a simple fictitious domain method with L2-penalty for el-liptic and parabolic problems. A priori estimates and theerror estimates for penalization problems are carefully in-vestigated. Our methods can be applied not only to FEMbut to FVM. Numerical experiments are performed to con-firm the theoretical results.
Guanyu ZhouGraduate School of Mathematical Sciences,The University of [email protected]
Norikazu SaitoUniversity of TokyoGraduate School of Mathematical [email protected]
CP10
Higher Order Energy Preserving Schemes for aNonlinear Variational Wave Equation
We consider a nonlinear variational wave equation describ-ing the dynamics of the director field in a nematic liquidcrystal. This equation has an intrinsic dichotomy betweenwhat is called conservative and dissipative solutions. Stan-dard finite difference schemes will, generally speaking, givea dissipative solution. In this talk, we discuss how to con-struct higher order energy preserving numerical schemesfor the nonlinear variational wave equation using the dis-continuous Galerkin framework.
Peder AursandNorwegian University of Science and [email protected]
Siddhartha MishraETH ZurichZurich, [email protected]
Ujjwal KoleyUniversity of [email protected]
CP10
TravelingWaves of Moderate Amplitude in ShallowWater
We prove existence of traveling wave solutions of an equa-tion for surface waves of moderate amplitude arising asa shallow water approximation of the Euler equations forinviscid, incompressible and homogenous fluids. Our ap-proach is based on techniques from dynamical systems andrelies on a reformulation of the equation as an autonomousHamiltonian system. We determine bounded orbits in thephase plane to establish existence of the corresponding pe-riodic and solitary traveling wave solutions of elevation anddepression, as well as of solitary waves with compact sup-port. Furthermore, we provide a detailed analysis of thesolitary waves’ dependence on the wave speed.
Anna GeyerUniversity of [email protected]
CP10
On-Site and Off-Site Solitary Waves of the Discrete
PD13 Abstracts 67
Nonlinear Schrodinger Equation
We construct several types of symmetric localized stand-ing waves (solitons) to the d-dimensional discrete nonlinearSchrodinger equation (dNLS) with cubic nonlinearity ford = 1, 2, 3 : i∂tun = h−2(δ2u)n−|un|2un, where δ
2 denotes
the discrete Laplacian on Zd, using bifurcation methods
from the continuum limit. Such waves and their relativeenergies are thought to play a role in the propagation of lo-calized states of dNLS across the lattice, which radiate en-ergy until they converge to a solitary wave at a fixed latticesite. This phenomenon is known as the ”Peierls-NabarroBarrier” in discrete systems and was first investigated for-mally by M. Peyrard and M.D. Kruskal. These standingwave solutions and the question of their stability proper-ties are also closely related to the variational problem ofseeking critical points of the Hamiltonian subject to fixedl2 norm (ground states).
Michael JenkinsonColumbia UniversityDepartment of Applied Physics and Applied [email protected]
Michael I. WeinsteinColumbia UniversityDept of Applied Physics & Applied [email protected]
CP10
On a Nonlinear Dispersive Wave of Third Order
We discuss several features of a nonlinear modulated dis-persive wave of third order. We show that under certainconditions, the localized smooth solution decays in time.The rate of the decay is also obtained.
Jeng-Eng LinGeorge Mason [email protected]
CP10
Shock-Wave Analysis of Condensate Developmentin a Model of Photon Scattering
We study long-time dynamics in a simplified Kompaneetsequation. The Kompaneets equation describes evolutionof photon energy spectrum due to Compton scattering ofphotons by electrons, an important energy transport mech-anism in certain plasmas. For our model, we prove globalexistence for initial data with any finite moment, conver-gence to equilibrium in large time, and failure to conservephoton number for large solutions, due to formation of ashock at zero energy. This is joint work with Dave Lever-more and Hailiang Liu.
Robert PegoCarnegie Mellon [email protected]
CP10
Relating Collision-Induced Dynamics of Soliton Se-quences of Coupled-Nls Equations and Dynamics inLotka-Volterra Models
We show that collision-induced dynamics of sequences ofsolitons of N perturbed coupled nonlinear Schrodinger(NLS) equations can be described by N dimensional Lotka-
Volterra models, and that the form of the LV model de-pends on the nature of the perturbation term. Stabilityand bifurcation analysis for the equilibrium points of theLV models is used to stabilize soliton-sequence propagationand to achieve on-off switching. Furthermore, for some se-tups, the LV description appears to hold even outside ofthe perturbative regime.
Avner PelegState University of New York at [email protected]
Debananda ChakrabortyVirginia Intermonte [email protected]
Quan M. NguyenInternational University, Vietnam [email protected]
Yeojin ChungSouthern Methodist [email protected]
Jae-Hun JungSUNY at [email protected]
CP10
On Models of Short Pulse Type in Continuous Me-dia
We consider ultra-short pulse propagation in nonlinearmetamaterials characterized by a weak Kerr-type nonlin-earity in their dielectric response. We will derive short-pulse equation (SPE) uxt = u + 1
6(u3)xx in frequency
band gaps. We will discuss SPE in its characteristiccoordinate and the transformation to sine-Gordon equa-tion, and robustness of various solutions emanating fromthe sine-Gordon equation and their periodic generaliza-tions. By adding a regularized term in the equation,we will discuss the connection of (regularized) SPE withthe nonlinear Schrodinger equation. Finally we will dis-cuss the wellposedness of generalized Ostrovsky equationuxt = u+ 1
6(up)xx with small initial data.
Yannan ShenUniversity of [email protected]
CP11
Nonlocal Weighted Biological Aggregation
The Cucker-Smale flocking model has a well-knownAchilles heel: the diluting effect of distant mass. A simi-lar weakness afflicts typical nonlocal aggregation equationswith attractive-repulsive interaction. To address the lat-ter we consider a weighted-averaging approach similar toMotsch-Tadmor’s adjustment of Cucker-Smale. We shallexamine how gradient-flow structure is maintained via theintroduction of a new metric, one that penalizes movementin crowded configurations (nonlocally). We will interpretthe metric, consider its pairing with several related energypotentials under gradient flow, and discuss its influence onaggregation dynamics.
Jeff Eisenbeis, Robert Pego, Dejan Slepcev
68 PD13 Abstracts
Carnegie Mellon [email protected], [email protected],[email protected]
CP11
Front-Dynamics and Pattern Selection in Semi-Bounded Domains: Pulled Vs Pushed Fronts inCahn-Hilliard, CGL, and FHN
Invasion fronts govern pattern-forming processes in manyexperimental areas such as ion bombardment, chemical nu-cleation, and phase separation. Often, an instability is trig-gered spatially progressively, creating an effective bound-ary condition ahead of an invasion front. We describe theeffect of the boundary condition on the front dynamics andthe patterns in the wake. We find two basic scenarios cor-responding to pushed and pulled fronts. Pushed fronts leadto oscillatory nonlinear interaction, hysteresis, and multi-stability in the presence of boundaries. Pulled fronts in-teract monotonically, with universal leading-order termsdetermined by absolute spectra. We prove these results, inparticular establishing a complete bifurcation diagram andleagding-order asymptotics, in the complex Ginzburg Lan-dau equation using geometric desingularization and hete-roclinic matching. We also present numerical studies inCahn-Hilliard and FitzHugh-Nagumo equations. Joint w/A Scheel.
Ryan GohUniversity of [email protected]
CP11
Multi-Agent Control of the Generalized ViscousBurgers Equation
We model the dynamic properties of traffic flows by meansof the generalized viscous Burgers equation defined overfinite space. Our aim is to determine the spatio-temporaldistribution of adaptive control agents for tracking the for-mation of traffic density patterns and for distributing thecontrol feedback throughout the spatial domain. Feedbackcontrol is then applied to stabilize the dynamics of theBurgers equation to maximize the traffic flows or triggerfinite-time transition along equilibrium profiles of the gov-erning PDE.
Dimitrios [email protected]
CP11
Bi-Stable Mean-Field Model
In this poster, I will present the bi-stable mean-field modelto illustrate the rich long-time dynamics of stochastic mod-els. The mathematical analysis of this model, including thestudy of mean-field limits, existence of multiple equilibria,and fluctuation theory was initiated by Dawson et al. Us-ing the large-deviations theory Papanicolaou et al. recentlystudied the dynamical phase-transitions in such systemsdue to stochastic fluctuation. We consider various gener-alizations of such mean-field models and study the phasetransitions between different equilibria in various settings.
Chao TianPennsylvania State Universitytian [email protected]
Qiang DuPenn State UniversityDepartment of [email protected]
CP11
Different Wave Solutions Associated with SingularLines on Phase Plane
The bifurcation phenomena of a compound K(m,m) equa-tion are investigated in this paper. Three singular lineshave been found in the associated topological vector field,which may evolve in the phase trajectories of the system.The influence of parameters as well as the singular lines onthe properties of the equilibrium points has been exploredin details. Transition boundaries have been obtained todivide the parameter space into regions associated withdifferent types of phase trajectories. The existence con-ditions and related discussions for different traveling wavesolutions have been presented in the end.
Yu V. WangYork College, The City University of New [email protected]
CP12
Steklov Representations of Harmonic Functions onExterior Regions
This talk will summarize some results about the represen-tation of solutions of Laplace’s equation in exterior regionsin R3 subject to Dirichlet, Robin or Neumann boundarydata, The results generalize the theory of multipole expan-sions used for the region outside a ball. This is used todescribe some new formulae for the electrostatic capacityof bounded sets in space. This is joint work with Qi Han.
Giles AuchmutyDepartment of MathematicsUniversity of [email protected]
CP12
Differentiability of the Function Best Sobolev Con-stant
Let Ω be a bounded smooth domain of RN , 1 ≤ p ≤ Nand p� := Np
N−p. We consider the function q ∈ [1, p�] → λq,
where
λq := inf
{∫Ω
|∇u|p dx : u ∈ W 1,p0 (Ω) and
∫Ω
|u|q dx = 1
}.
We prove that this function is absolutely continuous andalso given by the expression
λq = λ1 exp
(−p
∫ q
1
1
s
∫Ω
|us|s log |us| dxds)
for all q ∈ [1, p�], where
us ∈ Es :=
{u ∈ W 1,p
0 (Ω) :
∫Ω
|u|s dx = 1 and
∫Ω
|∇u|p dx = λs
}
for each s ∈ [1, p�). As a consequence, λq is differentiableat q ∈ [1, p�) if, and only if, the functional
Iq(u) :=
∫Ω
|uq|q log |uq | dx
PD13 Abstracts 69
is constant on Eq. It follows as an application of this resultthat the function q → λq belongs to C1 ([1, p�)) if Ω isa ball. Moreover, since q → λq is differentiable almosteverywhere we obtain the following new observation: foralmost all q ∈ (p, p�) the functional Iq is constant on theset Eq.
Grey ErcoleUFMG - UNIVERSIDADE FEDERAL DE [email protected]
CP12
Bubbling Solutions for the Chern-Simons GaugedO(3) Sigma Model
We construct multivortex solutions of the elliptic govern-ing equation for the self-dual Chern-Simons gauged O(3)sigma model. Our solutions show concentration phenom-ena at some points of the singular points as the couplingparameter tends to zero, and they are locally asymptoti-cally radial near each blow-up point.
Jongmin HanKyung Hee [email protected]
Kwangseok ChoeInha [email protected]
Chang-Shou LinNational Taiwan [email protected]
CP12
A Semismooth Newton Multigrid Method for Con-strainedElliptic Optimal Control Problems
A multigrid scheme is proposed for solving the Schur com-plement linear systems arising in each Newton iterationwhen the semi-smooth Newton method is applied to solvecontrol-constrained elliptic optimal control problems. Nu-merical experiments are performed to illustrate the highefficiency of our proposed method. Computation simula-tion show that the convergence rate is quite robust as theregularization parameter approaches to zero.
Jun LiuSouthern Illinois [email protected]
Mingqing XiaoSouthern Illinois University at [email protected]
CP12
A Free Boundary Problem for Higher Order Ellip-tic Operators
Let Ω be a domain in Rn with 0 ∈ ∂Ω. Suppose in B, the
unit ball in Rn, u and Ω satisfy the following equation in
the sense of distributions:
Lu = χΩ in B
nonumber Dαu = 0 for |α| ≤ 3 in B \ Ω.
Here L is a homogeneous fourth order elliptic operator andχΩ denotes the characteristic function. We analyze theregularity properties of u.
Henok MawiHoward [email protected]
CP12
Numerical Solution of Nonlinear Elliptic EquationUsing Finite Element Method
In this paper we have discussed the behavior, difficultiesand limitations involved in solving fully nonlinear ellipticpartial differential equations. Then we worked on a finiteelement method based on strong form of the differentialequation which works well for fully nonlinear elliptic equa-tion without much assumptions and illustrated numericalresult with the help of MATLAB.
Garima MishraResearch ScholarResearch [email protected]
Manoj KumarMotilal Nehru National Institute of TechnologyAllahabad, [email protected]
CP13
Semicontinuous Viscosity Solutions for Quasicon-vex Hamiltonians
The theory of semicontinuous viscosity solutions for con-vex hamiltonians introduced by Barron and Jensen in 1990is extended to quasiconvex hamiltonians. Equations of theform H(x, u(x), Du(x)) = 0, with p → H(x, r, p) quasi-convex, are shown to have a unique lower semicontinuousviscosity solution and such a solution is represented as thevalue function to a variational problem in L∞.
Emmanuel BarronLoyola University [email protected]
CP13
A Hamilton-Jacobi Equation for the ContinuumLimit of Non-Dominated Sorting
Non-dominated sorting is a fundamental problem in multi-objective optimization, and is equivalent to several impor-tant combinatorial problems. The sorting can be viewedas arranging points in Euclidean space into fronts by re-peated removal of the set of minimal elements with respectto the componentwise partial order. We prove that in the(random) large sample size limit, the fronts converge al-most surely to the level sets of the viscosity solution of aHamilton-Jacobi equation.
Jeff CalderDepartment of MathematicsUniversity of [email protected]
Selim EsedogluUniversity of Michigan
70 PD13 Abstracts
Alfred O. HeroThe University of MichiganDept of Elect Eng/Comp [email protected]
CP13
General Solution to Unidimensional Hamilton-Jacobi Equation
A method for finding the general solution to the par-tial differential equations: F (ux, uy) = 0;F (f(x)ux, uy) =0(orF (ux, h(y)uy) = 0) is presented, founded on a Leg-endre like transformation and a theorem for Pfaffian dif-ferential forms. As the solution obtained depends on anarbitrary function, then it is a general solution. As anextension of the method it is obtained a general solutionto PDE: F (f(x)ux, uy) = G(x),and then applied to unidi-mensional Hamilton-Jacobi equation.
Maria Lewtchuk EspindolaMathematics Department - Universidade Federal [email protected]
CP13
The Pullback Equation: An Overview
This talk will summarize some results I obtained with var-ious authors and led to the publication of a book: [Csato-Dacorogna-Kneuss,The Pullback Equation for DifferentialForms, Springer, 2012]. This equation consists of finding amap ϕ : Rn → Rn satisfying
ϕ∗(g) = f
where g and f are closed differential k-forms. This turnsout to be a nonlinear first-order system of
(nk
)partial differ-
ential equations. I will emphasis the cases k = 2 and k = nwhere we obtained global results with optimal regularityand Dirichlet condition.
Olivier KneussDepartment of MathematicsUniversity of California, [email protected]
CP14
Long-time Behavior of Solutions to the GeneralizedTwo-Component Hunter-Saxton System
We examine regularity of solutions to the non-local system
uxt+uuxx−a
2u2x−k
2ρ2 = −k
2
∫ 1
0
ρ2dx−a+ 2
2
∫ 1
0
u2xdx, (1)
ρt + uρx = aρux, (a, k) ∈ R2.
System (1) generalizes the Hunter-Saxton equation whichdescribes the orientation of waves in massive nematic liq-uid crystals. It also appears as the short-wave limit(a = −1, k = ±1) of the Camassa-Holm equation arisingin the theory of shallow water waves, the inviscid Karman-Batchelor flow (a = −k = 1), the Constantin-Lax-Majdaequation (a = −k = ∞), and the generalized Proudman-Johnson equation comprising several models from fluid dy-namics. Moreover, (1) serves as a tool for studying therole that one-dimensional convection and stretching play
in hydro-dynamically relevant evolution equations. In thispresentation a general solution formula to (1) is derivedand its regularity for a class of smooth initial data is stud-ied.
Alejandro SarriaUniversity of Colorado, BoulderPostdoctoral appointment to begin in Fall [email protected]
CP14
Boundary Value Problems of the System of Pdesof Steady Vibrations in the Theory of Thermoelas-ticity for Solids with Double Porosity
The boundary value problems of the system of PDEs ofsteady vibrations in the theory of thermoelasticity forsolids with double porosity are investigated. The funda-mental solution of this system is constructed by means ofelementary functions. The integral representations formu-las of classical solutions are presented. The basic propertiesof potentials are established. The uniqueness and existencetheorems for boundary value problems are proved by thepotential method and the theory of singular integral equa-tions.
Merab SvanadzeIlia State [email protected]
CP14
From Micropolar Navier-Stokes Equations to Fer-rofluids: Analysis and Numerics
The Micropolar Navier-Stokes Equations (MNSE), is a sys-tem of nonlinear parabolic partial differential equationscoupling linear velocity, pressure and angular velocity, i.e.:material particles have both translational and rotationaldegrees of freedom. The MNSE is a central component ofthe Rosensweig model for ferrofluids, describing the linearvelocity, angular velocity, and magnetization inside the fer-rofluids, while subject to distributed magnetic forces andtorques. We present the basic PDE results for the MNSE(energy estimates and existence theorems), together with afirst order semi-implicit fully-discrete scheme which decou-ples the computation of the linear and angular velocities.Similarly, for the Rosensweig model we present the basicPDE results, together with an fully-discrete scheme com-bining Continuous Galerkin and Discontinuous Galerkintechniques in order to guarantee discrete energy stability.Finally, we demonstrate the capabilities of the Rosensweigmodel and its numerical implementation with some numer-ical simulations in the context of ferrofluid pumping bymeans of external magnetic fields.
Ignacio TomasUniversity of Maryland, College [email protected]
CP14
Damping by Heat Conduction in the TimoshenkoSystem: Fourier and Cattaneo Are the Same
We consider the Cauchy problem of the one-dimensionalTimoshenko system coupled with heat conduction, whereinthe latter is described by either the Cattaneo law or theFourier law. We prove that heat dissipation alone is suf-ficient to stabilize the system in both cases, so that ad-ditional mechanical damping is unnecessary. However, the
PD13 Abstracts 71
decay of solutions without the mechanical damping is foundto be slower than that with mechanical damping. Fur-thermore, in contrast to earlier results, we find that theTimoshenko-Fourier and the Timoshenko-Cattaneo sys-tems have the same decay rate. The rate depends on acertain number, which is a function of the parameters ofthe system.
Aslan R. KasimovDivision of Mathematical & Computer Sciences andEngineeringKing Abdullah University of Science and [email protected]
belkacem said-houariking abdullah university of science and [email protected]
MS1
Concerning the Rational Decay of Certain Fluid-Structure PDE Models
In this talk, we shall demonstrate how delicate frequencydomain relations and estimates, associated with coupledsystems of partial differential equation models (PDE’s),may be exploited so as to establish results of uniform andrational decay. In particular, our focus will be upon decayproperties of coupled PDE systems of different character-istics; e.g., hyperbolic versus parabolic characteristics. Forsuch PDE systems of contrasting dynamics, the attainmentof explicit decay rates is known to be a difficult problem,inasmuch as there has not been an established methodol-ogy to handle hyperbolic-parabolic systems. For uncoupledwave equations or uncoupled heat equations, there are spe-cific Carleman’s multiplier methods in the time domain,wherein the exponential weights in each Carleman’s mul-tiplier carefully take into account the particular dynamicsinvolved, be it hyperbolic or parabolic. But for coupledPDE systems which involve hyperbolic dynamics interact-ing with parabolic dynamics, typically across some bound-ary interface, Carleman’s multipliers are readily applicable.Given that such coupled PDE systems occur frequently innature and in engineering applications; e.g., fluid-structureand structural acoustic interactions, there is a patent needto devise broadly implementable techniques by which onecan infer uniform decay for a given PDE system. As oneparticular example, we shall work to conclude uniform de-cays for structural acoustic dynamics. In these PDE mod-els, the structural component is subjected to a structuraldamping ranging from viscous (weak) to strong (Kelvin-Voight). The rational decay rates we derive for this prob-lem explicitly reflect the extent of the damping which is inplay. Since the damped elastic component of the coupleddynamics is present on only a portion of the boundary,there will necessarily be assumptions imposed upon thegeometry.
George AvalosUniversity of Nebraska-LincolnDepartment of Mathematics and [email protected]
MS1
Flutter of a Cantilever Beam in Low Speed AxialAir Flow:Theory and Experi Ment
Abstract not available at time of publication.
A.V. Balakrishnan
University of California Los [email protected]
MS1
On a Free Boundary Fluid-Elasticity Interaction
The talk addresses the problem of minimizing turbulenceinside fluid flow in the case of free boundary interaction be-tween a viscous fluid and a moving and deforming elasticbody. Our problem is motivated by the issue of reducingturbulence inside the blood flow in stenosed and stentedarteries. The presentation will focus on (i) existence of anoptimal control acting inside the fluid domain; and (ii) thefirst-order necessary conditions of optimality, which involvefinding a suitable adjoint problem and using it to explic-itly compute the gradient of the cost functional. This willprovide the characterization of the optimal control, pavingthe way for a numerical study of the problem.
Lorena BociuNorth Carolina State [email protected]
MS1
Fluid-Structure Interaction with Multiple Struc-tural Layers
We study a nonlinear, unsteady, moving boundary, fluid-structure (FSI) problem in which the structure is com-posed of two layers: a thin layer which is in contact withthe fluid, and a thick layer which sits on top of the thinstructural layer. The fluid flow, which is driven by thetime-dependent dynamic pressure data, is governed by the2D Navier-Stokes equations for an incompressible, viscousfluid, defined on a 2D cylinder. The elastodynamics of thecylinder wall is governed by the 1D linear wave equationmodeling the thin structural layer, and by the 2D equationsof linear elasticity modeling the thick structural layer. Weprove existence of a weak solution to this nonlinear FSIproblem as long as the cylinder radius is greater than zero.The spaces of weak solutions reveal a striking new feature:the presence of a thin fluid-structure interface with massregularizes solutions of the coupled problem.
Suncica CanicDepartment of MathematicsUniversity of [email protected]
Boris MuhaDepartment of Mathematics, Faculty of ScienceUniversity of [email protected]
Martina BukacUniversity of PittsburghDepartment of [email protected]
MS2
The Stochastic Navier-Stokes Equation and theStatistical Theory of Turbulence
In 1941 Kolmogorov and Obukhov proposed that there ex-ists a statistical theory of turbulence that allows the com-putation of all the statistical quantities that can be com-puted and measured in turbulent systems. In this talk,we will outline how to construct the Kolmogorov-Obukhov
72 PD13 Abstracts
refined scaling hypothesis (1962) with the She-Leveque in-termittency corrections from the stochastic Navier-Stokes(SNS) equation. We first estimate the structure functionsof turbulence and establish the Kolmogorov-Obukhov ’62scaling hypothesis with the She-Leveque intermittency cor-rections. Then we compute the invariant measure of tur-bulence writing the stochastic Navier-Stokes equation asan infinite-dimensional Ito process and solving the linearKolmogorov-Hopf functional differential equation for theinvariant measure.
Bjorn BirnirUniversity of CaliforniaDepartment of [email protected]
MS2
Optimal Stirring and Maximal Mixing
Passive advection of a tracer by an incompressible flowfield typically results in mixing of the scalar field, an im-portant phenomena for applications in science and engi-neering. Mixing may be measured via suitable norms ofthe tracer density, allowing for the formulation of sensiblequestions regarding optimal stirring strategies. In this talkwe discuss current research aimed at articulating effectiveestimates of maximal mixing rates for specific classes ofstirring flows.
Charles R. DoeringUniversity of MichiganMathematics, Physics and Complex [email protected]
MS2
Determining Form for the 2D Navier-Stokes Equa-tions Via Feedback Control.
Data assimilation uses a finite dimensional trajectories ina feedback control term to determine the full solution tothe 2D Navier-Stokes equations. The data can be a gen-eral interpolant such as finite volume elements, nodal val-ues, or Fourier modes. We discuss the evolution of thefinite dimensional data itself, through an ordinary differ-ential equation on a set of bounded trajectories. This is analternative to an earlier approach which works for Fouriermodes.
Ciprian FoiasTexas A & M,Indiana [email protected]
Michael S. JollyIndiana UniversityDepartment of [email protected]
Rostyslav KravchenkoUniversity of [email protected]
Edriss S. TitiUniversity of California, Irvine andWeizmann Institute of Science, Israel
[email protected], [email protected]
MS2
Ergodic Control for Stochastic Navier-StokesEquations
After a brief introduction to solvability and ergodic behav-ior of stochastic Navier-Stokes equations, the existence ofoptimal ergodic controls will be established for the two-dimensional stochastic Navier-Stokes equation. A con-trolled martingale problem with relaxed controls will pro-vide the framework for this study. Further properties ofthe optimal control will be discussed.
P. SundarLousiana State [email protected]
S.S. SritharanNaval Postgraduate [email protected]
MS3
Spectral Stability for Transition Fronts in Multidi-mensional Cahn-Hilliard Systems
We consider the spectrum associated with the linear oper-ator obtained when a Cahn-Hilliard system on R
n is lin-earized about a planar transition front solution. In thecase of a single Cahn-Hilliard equation on R
n, it’s knownthat under general physical conditions the leading eigen-value moves into the negative real half plane at a rate |ξ|3,where ξ denotes the Fourier variable corresponding withcomponents transverse to the wave. Moreover, in the caseof a single equation, it has recently been verified that thisspectral stability implies nonlinear stability. In the currentanalysis, we establish a framework for verifying that thespectrum for multidimensional systems follows this samecubic law, and we use this framework to show that thecondition holds for certain example cases.
Peter HowardDepartment of MathematicsTexas A&M [email protected]
MS3
Morphological Competition in Amphiphilic Sys-tems
We present an analysis of the bifurcation and competitiveevolution of co-existing morphological structures of dis-tinct co-dimension as supported in the gradient flows ofthe Functionalized Cahn-Hilliard (FCH) free energy. Thesesystems support bilayer, pore, and micelle structures, aswell as complex network formed from the union of suchstructures. We analyse the pearling and meander instabil-ities of bilayer structures, and derive sharp-interface limitsfor the co-evolution of bilayer (homoclinic co-dimension 1)and pore (radial co-dimension 2) structures, showing thatpossibility of co-existence.
Keith PromislowMichigan State University
PD13 Abstracts 73
MS3
Rigorous Computation of Connecting Orbits
Connecting orbits play an organizing role in dynamicalsystems described by partial differential equations. Theydescribe localized patterns (pulses, boundary layers), actas building blocks for more complicated patterns, and de-scribe the transitions between different states. The pastdecade has seen enormous advances in the development ofcomputer assisted proofs in dynamics. In this talks we willdiscuss recent progress in the rigorous computation of con-necting orbits, and directions for further research will beoutlined.
Jan Bouwe Van Den BergVU University AmsterdamDepartment of [email protected]
Jean-Philippe LessardUniversite [email protected]
MS3
The Stability of Periodic Patterns of LocalizedSpots for Reaction-Diffusion Systems in R2
We determine the spectral stability threshold for a peri-odic arrangment of localized spots for some singularly per-turbed two-component reaction-diffusion systems includ-ing the Gierer-Meinhardt, Schnakenburg, and Gray-Scottmodels. In the semi-strong interaction asymptotic limitwhere only one of the components has an asymptoticallysmall diffusivity, the well-known leading order stabilitythreshold governing amplitude instabilities is independentof the lattice structure. By combining a spectral approachbased on Floquet-Bloch theory together with the method ofmatched asymptotic expansions and appropriate Fredholmsolvability condition, the next order term in the expan-sion of the stability threshold is calculated in terms of theregular parts of certain Green’s functions. From an opti-mization of this term, the most stable lattice arrangementof spots is identified.
Michael WardDepartment of MathematicsUniversity of British [email protected]
David IronDalhousie University, [email protected]
John RumseyDalhousie [email protected]
Juncheng WeiDepartment of MathematicsChinese University of Hong [email protected]
MS4
Higher order O(N) schemes for hyperbolic prob-
lems using successive convolution
We present recent results for obtaining higher order, A-stable schemes for the wave equation. The method relieson approximating derivatives with successive applicationsof a certain convolution integral. The convolution stabilizeshigher order schemes, and furthermore can be applied inO(N) operations using fast convolution techniques. We alsobriefly discuss extending the methodology to first orderhyperbolic conservation laws.
Matthew F. CausleyMichigan State [email protected]
Andrew J. ChristliebMichigan State UniverityDepartment of [email protected]
MS4
A New Discontinuous Galerkin Finite ElementMethod for Directly Solving the Hamilton-JacobiEquations
In this talk, we will introduce a new discontinuous Galerkinmethod for solving time-dependent Hamilton-Jacobi equa-tions. This new method works directly on the solutions,and makes use of a new entropy fix inspired by Harten andHyman’s entropy fix for Roe scheme for the conservationlaws. Compared to previous works, the method has theadvantage of simplicity in implementations, and could beapplied to compute problems with non convex Hamiltoni-ans.
Yingda Cheng, Zixuan WangDepartment of MathematicsMichigan State [email protected], [email protected]
MS4
Optimal energy conserving local discontinuousGalerkin methods for second-order wave equationin heterogeneous media
Solving wave propagation problems within heterogeneousmedia has been of great interest and has a wide range ofapplications in physics and engineering. The design of nu-merical methods for such general wave propagation prob-lems is challenging because the energy conserving propertyhas to be incorporated in the numerical algorithms in or-der to minimize the phase or shape errors after long timeintegration. In this talk, we will discuss multi-dimensionalwave problems and consider linear second-order wave equa-tion in heterogeneous media. We will present an LDGmethod, in which numerical fluxes are carefully designedto maintain the energy preserving property and accuracy.We propose compatible high order energy conserving timeintegrators and prove the optimal error estimates and theenergy conserving property for the semi-discrete methods.Our numerical experiments demonstrate optimal rates ofconvergence, and show that the errors of the numerical so-lution do not grow significantly in time due to the energyconserving property.
Ching-Shan ChouDepartment of MathematicsOhio State [email protected]
74 PD13 Abstracts
Yulong XingDepartment of MathematicsUniveristy of Tennessee / Oak Ridge National [email protected]
Chi-Wang ShuBrown UniversityDiv of Applied [email protected]
MS5
Multiple Solutions to An Elliptic Problem Relatedto Vortex Pairs
Abstract not available at time of publication.
Yi LiWright State UniversityXian Jiaotong [email protected]
MS5
Front Propagation in Isothermal Diffsuion Systems
Abstract not available at time of publication.
Yuanwei QiUniversity of Central [email protected]
MS5
Global Dynamical Analysis of An Age-StructuredCholera Model
Abstract not available at time of publication.
Zhisheng ShuaiUniversity of Central [email protected]
MS5
A Nonlinear Problem from the Flow Between TwoCounter-Rotating Disks
Abstract not available at time of publication.
weiqing XieCal Poly [email protected]
MS6
A Forward-backward Regularization of the Perona-Malik Equation
In this talk we will survey recent developments in the anal-ysis of the Perona-Malik equation as well as related modelswith particular emphasis on a forward-backward regular-ization which turns out to be well-posed in the sense ofweak Young measure solutions.
Patrick GuidottiUniversity of California at [email protected]
MS6
Numerical Implementation of a New Class of
Forward-backward-forward Diffusion Equations forImage Restoration
In this talk, we present the implementation and numer-ical experiments demonstrating new forward-backward-forward nonlinear diffusion equations for noise reductionand deblurring, developed in collaboration with PatrickGuidotti and Yunho Kim. The new models preserve andenhance the most desirable aspects of the closely-relatedPerona-Malik equation without allowing staircasing. Byusing a Krylov subspace spectral (KSS) method for time-stepping, the properties of the new models are preservedwithout sacrificing efficiency.
James V. LambersUniversity of Southern MississippiDepartment of [email protected]
Patrick GuidottiUniversity of California at [email protected]
Yunho KimYale [email protected]
MS6
Denoising an Image by Denoising its Curvature Im-age
In this work we argue that when an image is corrupted byadditive noise, its curvature image is less affected by it,i.e. the PSNR of the curvature image is larger. We specu-late that, given a denoising method, we may obtain betterresults by applying it to the curvature image and then re-constructing from it a clean image, rather than denoisingthe original image directly. Numerical experiments con-firm this for several PDE-based and patch-based denoisingalgorithms.
Stacey LevineDuquesne [email protected]
Marcelo BertalmioUniversitat Pompeu [email protected]
MS6
The Influence of Pde Based Image Processing inAnalysis: Modern Developments
Abstract not available at time of publication.
Surya PrasathUniversity of [email protected]
MS7
Overturning Traveling Waves in Interfacial FluidDynamics
Large-amplitude traveling waves in interfacial fluid dynam-ics can, in certain settings, be overhanging. In work withBenjamin Akers and J. Douglas Wright, we introduce a newformulation for the traveling wave problem which allows forsuch waves. This formulation gives a traveling wave ansatz
PD13 Abstracts 75
for a parameterized curve, making use of a normalized ar-clength parameterization. We apply this formulation, bothanalytically and computationally, to interfacial flows withsurface tension.
David AmbroseDrexel [email protected]
MS7
Low Regularity Well-posedness for the 2DMaxwell-Klein-Gordon Equation
We discuss progress on the low regularity local well-posedness for the Maxwell-Klein-Gordon equation in twodimensions.
Magda CzubakBinghamton [email protected]
MS7
The Discrete Schrodinger Equation on TriangularLattices
The discrete Schrodinger equation on a cubic lattice (inany number of dimensions) is easily solved by separationof variables. On a two-dimensional triangular lattice, how-ever, separation of variables is not possible. The funda-mental solution decays (in supremum norm) at the rate
|t|−3/4 for a generic choice of coupling coefficients betweenadjacent vertices. Slower power-law decay rates are ob-served if these coefficients satisfy a certain set of algebraicrelations.
Vita Borovyk, Michael GoldbergUniversity of [email protected], [email protected]
MS7
Well-Posedness Results for a Derivative NonlinearSchrodinger Equation
In this talk, I will discuss new results on the existence anduniqueness of solutions to a generalized derivative nonlin-ear Schrodinger equation. This equation, which has nonlin-earity i|u|2σux has recently been given some attention forits solitary wave solutions and blow up dynamics. How-ever, there are interesting values of σ, including 1 < σ ≤ 2,for which no existence or uniqueness results are available.Open problems will be highlighted.
Gideon SimpsonSchool of MathematicsUniversity of [email protected]
MS8
Global Well-Posedness of Boussinesq Equations
Boussinesq equations are mathematical models of buoy-ancy driven flows. In this talk we first introduce the Boussi-nesq equations, then, we will study the global in time exis-tence of classical solutions to some reduced 3D Boussinesqequations and also the 2D Boussinesq equations with par-tial dissipation.
Chongsheng Cao
Department of MathematicsFlorida International [email protected]
MS8
On the Navier-Stokes Limit of the Navier-Stokes-Voigt Equations
We consider the three-dimensional Navier-Stokes-Voigt(NSV) equations and we analyze, from the asymptoticbehavior view point, its Navier-Stokes (NS) limit as therelaxation parameter vanishes. We show that the NSV-attractors converge to the weak NS-attractor in the Haus-dorff semidistance induced by the weak L2-metric on theabsorbing set of the Navier-Stokes equations. Some resultsrelated to the strong topology of L2 are also proved.
Michele Coti ZelatiIndiana University, [email protected]
MS8
Vortex Stretching and Sub-Criticality for the 3DNavier-Stokes Equations
The goal of this talk is to present a new approach to theproblem of possible singularity formation in the 3D NSE.The first step consists of elucidating a geometric scenarioleading to criticality for large data. This is based on alocal, geometric measure-type regularity criterion, essen-tially, local, anisotropic one-dimensional sparseness of theregions of intense vorticity, coupled with the mathematicalevidence that the scale of sparseness needed to prevent theblow-up may in fact be achieved via the mechanism of vor-tex stretching. The second step consists of attempting tobrake the criticality (in the aforementioned sense/setting).Presently, this is indeed possible for a quite general classof blow-ups; in particular, for any blow-up rate featuringlocal spatially-algebraic structure.
Zoran GrujicUniversity of [email protected]
MS8
The Navier-Stokes-Voight Model for Image In-painting
We investigate some of the advantages of the 2D Navier-Stokes-Voight (NSV) turbulence model and explore its lim-its in the context of image inpainting. We begin by givinga brief review of the elegant analogy between the image in-tensity function for the image inpainting problem and thestream function in 2D incompressible fluid, first introducedby Bertalmio et al in 2001. We show that the NSV allowsfor a more efficient numerical algorithm when automatingthe inpainting process.
Moe EbrahimiUniversity of California, San [email protected]
Michael HolstUC San DiegoDepartment of [email protected]
Evelyn Lunasin
76 PD13 Abstracts
US Naval [email protected]
MS9
Asymptotically Preserving Scheme for Some Ki-netic and Fluids Equation with Multi-Scale Behav-iors
I will present some Asymptotically Preserving schemes insome kinetic and fluids equations, including low Mach flu-ids, a semilinear hyperbolic relaxation system with a two-scale discontinuous relaxation rate, linear kinetic equationsin the diffusion limit, kinetic-fluid modeling of disperse two-phase flows, etc. I will emphasis on numerical analysis andalgorithm designs for these singular and stiff problems.
Jian-guo LiuDept. of Mathematics and PhysicsDuke [email protected]
MS9
The Unified Gas Kinetic Scheme of K. Xu Appliedto Linear Transport in Diffusion Regimes
I will show that the unified gas kinetic scheme (UGKS) ofK. Xu et al., originally developed for gas dynamics, can beapplied to a linear kinetic model of radiative transfer the-ory. I will show that UGKS is asymptotic preserving (AP)in both optically thick and thin regimes. Contrary to manyAP schemes, this method is based on a standard finite vol-ume approach, it does neither use any decomposition ofthe solution, nor staggered grids.
Luc MieussensInstitut de Mathematiques de [email protected]
MS9
High Order Semi-Implicit Runge-Kutta Finite Dif-ference Methods and Application to Non-LinearParabolic Relaxation Problems
We study solutions to nonlinear hyperbolic systems withfully nonlinear relaxation terms in so called diffusive limit.Suitable semi-implicit schemes based on IMEX-Runge-Kutta are constructed for the numerical solution of therelaxed limit equation. Another set of schemes is devisedfor relaxation systems, which are able to capture the limitdiffusive relaxation with no parabolic time step restriction.A new stability analisys of the schemes is proposed andapplied.
Giovanni RussoUniversity of CataniaDepartment of Mathematics and Computer [email protected]
Sebastiano BoscarinoUniversity of [email protected]
MS9
On the Efficiency of Asymptotic PreservingSchemes
We apply the concept of Asymptotic Preserving (AP)schemes[Jin 1999] to the linearized p-system and discretize
the resulting elliptic equation using continuous Finite El-ements. We analyze the consistency and compare thescheme numerically with methods such as Implicit Euler.Numerical results indicate that the AP method is indeedsuperior to more traditional methods. We discuss exten-sions to higher order of consistency.
Jochen SchutzIGPMRWTH [email protected]
MS10
Wave Fronts in a Model for Gasless Combustionwith Heat Loss
We consider a model of gasless combustion with heat loss,where the heat loss from the system to the environment isformulated according to Newtons law of cooling. The sys-tem of partial differential equations that describe evolutionof the temperature and remaining fuel contains two smallparameters, a diffusion coefficient for the fuel and a heatloss parameter. We use geometric singular perturbationtheory to show existence of traveling waves in this systemand then study their spectral and nonlinear stability.
Anna GhazaryanDepartment of MathematicsMiami [email protected]
MS10
Coherent Structures in a Model for Mussel-AlgaeInteraction
We consider a model for formation of mussel beds on softsediments. The model consists of coupled nonlinear pdesthat describe the interaction of mussel biomass on the sed-iment with algae in the water layer overlying the musselbed. Both the diffusion and the advection matrices in thesystem are singular. We use Geometric Singular Pertur-bation Theory to capture nonlinear mechanisms of patternand wave formation in this system.
Vahagn ManukianMiami University [email protected]
MS10
Threshold Phenomena for Symmetric DecreasingSolutions of Reaction-Diffusion Equations
We study the long time behavior of solutions of the Cauchyproblem for nonlinear reaction-diffusion equations in onespace dimension with the nonlinearity of bistable, ignitionor monostable type. We prove a one-to-one relation be-tween the long time behavior of the solution and the limitvalue of its energy for symmetric decreasing initial data inL2 under minimal assumptions on the nonlinearities. Theobtained relation allows to establish sharp threshold resultsbetween propagation and extinction for monotone familiesof initial data in the considered general setting.
Cyrill B. MuratovNew Jersey Institute of TechnologyDepartment of Mathematical Sciences
PD13 Abstracts 77
MS10
Traveling Waves in Coupled Reaction-DiffusionModels with Degenerate Source Terms
We consider a general system of coupled nonlinear diffusionequations that do not have isolated rest states. Such sys-tems arise in many different applications, e.g., propagationof epidemics, ion transport in cellular matter, and evapo-ration and condensation in porous media. We show thatthe degeneracy in the source terms implies that travelingwaves have a number of surprising properties that are notpresent for systems with nondegenerate source terms. Inparticular, we show that traveling wave trajectories con-necting two stable rest states can exist generically only fordiscrete wave speeds and that families of traveling waveswith a continuum of wave speeds cannot exist.
Jonathan J. WylieDepartment of MathematicsCity University of Hong Kong, Kowloon, Hong [email protected]
MS11
Dispersive Estimates for Schrodinger Operators inDimension Two
In this talk we will discuss L1 → L∞ dispersive estimatesfor Schrodinger operators in dimension two with real anddecaying potentials in the case when there are no spectralassumptions at the edge of the spectrum. In addition wewill describe logarithmically weighted dispersive estimateswith an integrable time decay rate at infinity.
Burak ErdoganUniversity of Illinois, [email protected]
William GreenRose-Hulman Institute of [email protected]
MS11
Relaxation of Exterior Wave Maps to HarmonicMaps
We establish relaxation of arbitrary, finite energy, one-equivariant wave maps in 3 space dimensions exterior tothe unit ball to the 3-sphere with a Dirichlet condition atr = 1, to the unique stationary harmonic map in its degreeclass. This settles a recent conjecture of Bizon, Chmaj,Maliborski who proposed this problem as a model in whichto study stable soliton resolution and who observed thisasymptotic behavior numerically. This is joint work withC. Kenig and W. Schlag.
Andrew [email protected]
MS11
A New Approach to Soliton Stability for the KdVEquation
In this work, we consider the KdV equation in the expo-nentially weighted spaces of Pego and Weinstein. We prove
local well-posedness of the perturbation (weighted and un-weighted) in the Bourgain X1,b space, allowing us to recre-ate the Pego-Weinstein result via iteration. By combiningthis result with the I-method, we expect ultimately to ob-tain soliton stability for KdV with initial data too roughto be in H1.
Brian Pigott, Sarah RaynorWake Forest [email protected], [email protected]
MS11
Talbot Effect for the Cubic Nonlinear SchrodingerEquation on the Torus
We study the evolution of the one dimensional periodicNLS equation with bounded variation data. For the linearevolution, it is known that for irrational times the solutionis a continuous, nowhere differentiable fractal-like curve.For rational times the solution is a linear combination offinitely many translates of the initial data. We prove thata similar phenomenon occurs in the case of the cubic NLSequation.
Nikolaos TzirakisUniversity of Illinois at [email protected]
MS12
A Geometric Approach for Mean Curvature Mo-tion in 2D
In this talk I will present a new approach for mean cur-vature motion in 2D, which explicitly connects the curveshortening from the differential geometry setting with thelevel set formulation, in the viscosity solutions theory.This provides an exact analytical framework for its nu-merical implementation, which has an important applica-tion in image processing. It runs on line on any image athttp://www.ipol.im/.
Adina CiomagaL.E. Dickson InstructorUniversity of [email protected]
MS12
The Thin One Phase-Problem
We describe the regularity of the free boundary for the thinone-phase problem, in which the free boundary occurs ona lower dimensional subspace. This problem appears as amodel of a one-phase free boundary problem in the contextof the fractional Laplacian.
Daniela De SilvaBarnard College,Columbia [email protected]
MS12
On Congested Crowd Motion
We investigate the relationship between Hele-Shaw evolu-tion with a drift and a transport equation with a drift po-tential, where the density is transported with a constrainton its maximum. When the drift potential is convex, the
78 PD13 Abstracts
crowd density is likely to aggregate. We show that theevolving patch satisfies a Hele-Shaw type equation.
Inwon KimDepartment of [email protected]
Yao YaoDepartment of MathematicsUniversity of [email protected]
Damon [email protected]
MS12
Numerical Methods for Geometric Elliptic PartialDifferential Equations
A few important geometric PDEs will be discussed whichcan be solved using a numerical method called Wide Stencilfinite difference schemes. Focusing on the Monge-Ampereequation, I show how naive schemes can work well forsmooth solutions, but break down in the singular case.There are several numerical schemes which fail to converge,or converge only in the case of smooth solutions. I willpresent a convergent solver which is fast, comparable tosolving the Laplace equation a few times.
Adam ObermanSimon Fraser [email protected]
MS13
Gevrey Regularity of Strongly Damped WaveEquations with Hyperbolic Dynamic BoundaryConditions
We consider a linear system of coupled PDEs which con-sists of a strongly damped wave equation on a boundeddomain with dynamic boundary conditions governed by an-other wave equation (via the Laplace-Beltrami operator).We show that the system generates a strongly continuoussemigroup T (t) which is analytic when strong damping ispresent on the boundary, otherwise of Gevrey class. Inboth cases the flow exhibits a regularizing effect on thedata. In particular, we prove quantitative time-smoothingestimates of the form ‖(d/dt)T (t)‖ ≤ C|t|−1 for the firstcase, ‖(d/dt)T (t)‖ ≤ C|t|−2 for the second.
Jameson Graber, Irena M. LasieckaUniversity of [email protected], [email protected]
MS13
Analysis of the Ericksen-Leslie System for LiquidCrystals
In this talk we consider the Ericksen-Leslie system describ-ing the nematic phase of liquid crystals. We discuss localand global wellposedness results for various subclasses ofthis system. Moreover, we present a thermodynamicallyconsistent version of the so called simplified model. This
is joint work with J. Pruss and K. Schade.
Matthias HieberTU Darmstadt, [email protected]
MS13
Feedback Stabilization of Boussinesq Equationsand Numerical Simulations
In this talk we present theoretical and numerical results fora feedback control problem defined by a thermal fluid. Theproblem is motivated by recent interest in designing andcontrolling energy efficient building systems. In particular,we show that it is possible to locally exponentially stabilizethe nonlinear Boussinesq equations by applying finite di-mensional Neumann/Robin type boundary controllers. Inthe end, a 2D problem is used to illustrate the ideas anddemonstrate the computational algorithms.
Weiwei HuUniversity of Southern [email protected]
MS13
Well-Posedness and Small Data Global Existencefor An Interface Damped Free Boundary Fluid-Structure Model
We address the well-posedness of a fluid-structure interac-tion model describing the motion of an elastic body im-mersed in an incompressible fluid. The fluid-structure sys-tem consists of the incompressible Navier-Stokes equationsand a damped linear wave equation coupled through trans-mission boundary conditions on the free moving interfaceseparating the elastic body and the fluid. We provide apriori estimates for the local-in-time existence of solutionsfor a class of initial data which also guarantees uniqueness.The global-in-time existence of solutions relies on a keyenergy inequality which in addition provides exponentialdecay of solutions of the system for given sufficiently smallinitial data.
Mihaela IgnatovaUniversity of California, [email protected]
Igor KukavicaUniversity of Southern [email protected]
Irena LasieckaUniversity of [email protected]
Amjad TuffahaThe Petroleum InstituteAbu Dhabi, [email protected]
MS14
Generalized Gevrey Norms with Applications toDissipative Equations
The regular solutions of the Navier-Stokes equations arewell-known to be analytic in both space and time vari-ables, even when one starts with non-analytic initial data.The space analyticity radius has an important physical in-
PD13 Abstracts 79
terpretation. It demarcates the length scale below whichthe viscous effect dominates the (nonlinear) inertial effect.Foias and Temam introduced an effective approach to esti-mate space analyticity radius via the use of Gevrey normswhich, since then, has become a standard tool for study-ing analyticity. We extend this approach to a class ofdissipative equations, including critical and super-criticalquasigeostrophic equations, where the dissipation opera-tor is a fractional Laplacian. This necessitates the useof sub-analytic Gevrey classes and ”generalized” Gevreynorms and development of certain commutator estimatesin Gevrey classes to exploit the cancellation properties ofthe equation. This is achieved via the Littlewood-Paley de-composition and the Bony paraproduct formula. Thoughnot essential for the Navier-Stokes equations, such commu-tator estimates become crucial for the critical and super-critical quasigeostrophic equations. Applications includelarge time decay of higher order derivatives.
Animikh BiswasUniversity of North Carolinaat [email protected]
MS14
Bounds on Energy and Enstrophy for the 3DNavier-Stokes-α and Leray-α Models
We construct semi-integral curves which bound the pro-jection of the global attractor of the Sabra Shell model inthe plane spanned by the energy and enstrophy for the 3Dregime and the plane spanned by the enstrophy and palin-strophy for the 2D regime. The semi-integral curves dividethe plane into regions having limited ranges for the direc-tions of the flow. This allows us to estimate the averagetime it would take a solution to burst into a region of largeenstrophy and palinstrophy, respectively.
Aseel FarhatIndiana University [email protected]
Michael S. JollyIndiana UniversityDepartment of [email protected]
Evelyn LunasinUnited States Naval [email protected]
MS14
On Inviscid Limits for the Stochastic Navier-StokesEquations and Related Systems
One of the original motivations for the development ofstochastic partial differential equations traces it’s originsto the study of turbulence. In particular, invariant mea-sures provide a canonical mathematical object connectingthe basic equations of fluid dynamics to the statistical prop-erties of turbulent flows. In this talk we discuss some recentresults concerning inviscid limits in this class of measuresfor the stochastic Navier-Stokes equations and other re-lated systems arising in geophysical and numerical settings.This is joint work with Peter Constantin, Vladimir Sverakand Vlad Vicol.
Nathan Glatt-HoltzIndiana University
MS14
Stability of Compressible Viscous Waves in a Mov-ing Domain
We establish the sharp nonlinear stability criteria for com-pressible viscous surface-internal waves for two barotropicfluids near the equilibrium. The stability is characterizedby the jump in the equilibrium density smaller than thecritical jump value. When the lower fluid is heavier thanthe upper fluid, we will establish the zero surface tensionlimit. This is joint work with Ian Tice and Yanjin Wang.
Juhi JangUniversity of California, Riverside, [email protected]
Ian TiceCarnegie Mellon [email protected]
Yanjin WangXiamen Universityyanjin [email protected]
MS15
Nonlinearity Saturation as a Singular Perturbationof the Nonlinear Schrodinger Equation
Saturation of the Kerr nonlinearity in the nonlinearSchrodinger equation can be regarded as a singular per-turbation which avoids the ill-posedness associated withthe self-focusing singularity. An asymptotic expansion ispresented which takes into account multiple scale behav-ior in both the longitudinal and transverse directions. Themechanism for interaction of solitary wave structures andan adjacent field is determined.
Karl GlasnerThe University of ArizonaDepartment of [email protected]
Jordan Allen-FlowersProgram in Applied MathUniversity of [email protected]
MS15
Spectra of Functionalized Operators Arising fromHypersurfaces
Functionalized energies, such as the Functionalized Cahn-Hilliard, model phase separation in amphiphilic systems,in which interface production is limited by competitionfor surfactant phase, which wets the interface. This isin contrast to classical phase-separating energies, such asthe Cahn-Hilliard, in which interfacial area is energeticallypenalized. In binary amphiphilic mixtures, interfaces arecharacterized not by single-layers, which separate domainsof phase A from those of phase B via a heteroclinic connec-tion, but by bilayers, which divide the domain of the domi-nant phase, A, via thin layers of phase B formed by homo-clinic connections. Evaluating the second variation of theFunctionalized energy at a bilayer interface yields a func-tionalized operator. We characterize the center-unstablespectra of functionalized operators and obtain resolvent es-
80 PD13 Abstracts
timates to the operators associated with gradient flows ofthe Functionalized energies. This is an essential step to arigorous reduction to a sharp-interface limit.
Gurgen HayrapetyanCarnegie Mellon [email protected]
Keith PromislowMichigan State [email protected]
MS15
Spots and Stripes in NLS-type Equations withNearly One-dimensional Potentials
We consider the existence of spots and stripes for a classof NLS-type equations in the presence of nearly one-dimensional localized potentials. Under suitable assump-tions on the potential, we construct various types of waveswhich are localized in the direction of the potential andhave single- or multi-hump, or periodic profile in the per-pendicular direction. The analysis relies upon a spatialdynamics formulation of the existence problem, togetherwith a center manifold reduction. This reduction proce-dure allows these waves to be realized as uni- or multi-pulsehomoclinic orbits, or periodic orbits in a reduced systemof ordinary differential equations.
Todd KapitulaCalvin [email protected]
Mariana HaragusUniversite de [email protected]
MS15
Stability of Line Solitons for the KP-II Equation
I will talk on nonlinear stability of line soliton solutionsfor the KP-II equation with respect to transverse pertur-bations that are exponentially localized as x → ∞. Thelocal amplitude and the phase shift of the crest of the linesolitons are described by a system of 1D wave equationswith diffraction terms and jumps of the local phase shift ofthe crest propagate in a finite speed toward y = ±∞.
Tetsu MizumachiFaculty of MathematicsKyushu University, [email protected]
MS16
Convected Scheme Solution of the Boltzmann-Poisson System with Spectrally Accurate Phase-Space Resolution
The Boltzmann-Poisson system describes the evolution ofmultiple species of charged particles that interact througha self-consistent electric field and various collisional pro-cesses. Semi-Lagrangian methods like the ConvectedScheme (CS) are often used to solve the transport terms,because of their ability to take large time-steps (no CFLlimit) and their low numerical diffusion. The CS is massconservative and positivity preserving, and was recently ex-tended to arbitrarily high order of accuracy in phase-space:the new scheme was applied to the Vlasov-Poisson system
on periodic domains, and validated against classical 1D-1Vtest-cases. Here we include a simple linear collision oper-ator (elastic scattering) and extend the model to 1D-2V.Then we study the coupled dynamics of electrons and ionsin a bounded domain.
Yaman GucluDepartment of MathematicsMichigan State [email protected]
MS16
A High Order Time Splitting Method Based onIntegral Deferred Correction for Semi-LagrangianVlasov Simulations
The dimensional splitting semi-Lagrangian methods withdifferent reconstruction/interpolation strategies have beenapplied to kinetic simulations in various settings. However,the temporal error is dominated by the splitting error. Inorder to have numerical algorithms that are high order bothin space and in time, we propose to use the integral deferredcorrection (IDC) framework to reduce the splitting error.The proposed algorithm is applied to the Vlasov-Poissonsystem, the guiding center model and incompressible flows.
Andrew ChristliebMichigan State [email protected]
Wei GuoUniversity of [email protected]
Maureen MortonKent State [email protected]
Jingmei QiuUniversity of [email protected]
MS16
High Order Operator Splitting Methods Based onan Integral Deferred Correction Framework
Integral deferred correction methods have been shown tobe an efficient way to achieve arbitrary high order accu-racy and possess good stability properties. In this paper,we construct high order operator splitting schemes usingthe integral deferred correction (IDC) procedure to solvetwo dimensional initial-boundary-value problems. Further-more, we present analysis which establishes that the IDCmethod can correct for both the splitting and numericalerrors, lifting the order of accuracy by r with each cor-rection, where r is the accuracy of the method used tosolved the correction equation. Numerical examples in 2Don linear and nonlinear problems, with periodic and Dirich-let boundary conditions, are presented to demonstrate thetheoretical results.
Andrew Christlieb, Yuan LiuDepartment of MathematicsMichigan State [email protected], [email protected]
Zhengfu XuMichigan Technological University
PD13 Abstracts 81
Dept. of Mathematical [email protected]
MS16
Analysis of Optimal Superconvergence of Discon-tinuous Galerkin Method
We study the superconvergence of the error for the discon-tinuous Galerkin method for linear conservation laws. Ifwe apply piecewise kth degree polynomials, the error be-tween the DG solution and the exact solution is (k+2)th or-der superconvergent at the downwind-biased Radau points.Moreover, the DG solution is (k+2)th order superconver-gent both for the cell averages and for a particular error.The idea is not based on Fourier analysis. Numerical exper-iments demonstrate the optimality of the superconvergencerate.
Yang YangBrown [email protected]
Chi-Wang ShuBrown UniversityDiv of Applied [email protected]
MS17
Reduced-Order Models for Nonlinear Systems withTime-Periodic Solutions
In this talk, we discuss the use of approximate Floquettransformations to develop reduced-order models for com-plex flows that exhibit periodic solutions. Our strategy in-corporates the proper orthogonal decomposition using sen-sitivity analysis and an optimal selection of subintervalsfor which to build snapshots. Results will include reduced-order models for Navier-Stokes and Boussinesq flows in-cluding vortex shedding and driven cavity problems.
Jeff BorggaardVirginia TechDepartment of [email protected]
MS17
Reduced-Order Modeling for Adjoint Equations
Abstract not available at time of publication.
Florian De VuystEcole Normale Superieure [email protected]
MS17
Reduced Order Model Based on Approximated LaxPairs
A reduced-order model algorithm, called ALP, is proposedto solve nonlinear evolution partial differential equations.It is based on approximations of Lax pairs. Contrary toother reduced-order methods, like Proper Orthogonal De-composition, the space where the solution is searched forevolves according to a dynamics specific to the problem. Itis therefore well-suited to solving problems with progres-sive waves or front propagation. Numerical examples areshown for the linear advection, KdV and FKPP equations,
in one and two dimensions.
Jean-Frederic Gerbeau, Damiano LombardiINRIA [email protected], [email protected]
MS17
Rectification Procedure for Improved Accuracy inReduced Basis Context
Reduced basis techniques, as other model order reductionapproaches use the fact that the solution of the problemwe are interested in lies in a low complexity space, that,e.g. can be characterized by a small Kolmogorov n-width.A typical application frame is the solution to some param-eter dependent family of PDE. In such cases, the proce-dure consists in computing accurately some solutions of the”low complexity space” and extrapolate in the parameterspace by some approach (EIM, GEIM, Galerkin, coloca-tion..) when the solutions is required at some other valuesof the parameter. In some applications, the reduced basissolution, for those parameters where an accurate solutionhas been computed is not equal to that approximation. Ana posteriori ”rectification” process can then be proposedand solve this deviation. It appears that the global solu-tion procedure is the much improved by this rectificationtool. We shall present a variety of cases where such a rec-tification is useful, explain the reason why it improves theaccuracy and show the efficiency of this post processingaction
Yvon MadayUniversite Pierre et Marie Curieand Brown [email protected]
MS18
A Nonlocal Vector Calculus with Applications toDiffusion and Mechanics
Abstract not available at time of publication.
Max GunzburgSchool of Computational ScienceFlorida State [email protected]
MS18
Quadrature Methods in Peridynamics
Peridynamics is a nonlocal extension of classical solid me-chanics suitable for modeling failure and fracture. Peridy-namic models require evaluation of an integral, performednumerically by quadrature over a mesh. The domain ofintegration in general does not align with element edges,meaning that one must integrate over fractions of an ele-ment. In this presentation, I survey previous approaches toquadrature in peridynamics, and present a new approachfor accurate 3D quadrature for peridynamics models.
Michael L. ParksSandia National [email protected]
MS18
Asymptotic Behavior for Solutions in Diffusion
82 PD13 Abstracts
Models with Peridynamic-Type Nonlocality
In this talk I will present some recent results regarding thelong time behavior of solutions to nonlocal wave equationswith damping. These equations could appear as models innonlocal diffusion, when one uses the Cattaneo-Vernotteflux to replace the Fourier law. Several methods suitablefor this investigation will be presented, some of which arereminiscent of the local setting, and I will also point outsome of the major challenges encountered in the nonlocalcase.
Petronela RaduUniversity of Nebraska-LincolnDepartment of [email protected]
MS18
Origin and Effect of Nonlocality in a Composite
The purpose of this talk is to demonstrate from simplemechanical arguments that nonlocality is a basic prop-erty of heterogeneous media. By deriving a model for thesmoothed displacement field in a simple composite, it isshown that nonlocality emerges naturally. The nonlocalmodel is put into the form of a peridynamic material model.Computations demonstrate improved agreement with ex-perimental data on composite laminates when nonlocalityis included.
Stewart SillingSandia National [email protected]
MS19
Well-posedness of the Linear KdV Equation on aReal Line
In this talk wellposedness of a linear KdV equation
∂tu+3∑
j=0
aj(x, t)∂jxu = f
is investigated. This equation is a model problem, whoseunderstanding has applications to linear and nonlinearKdV-type equations. The third derivative term ∂3
x makesthis equation dispersive and in the non-degenerate casea3 ≈ 1, the wellposedness is similar to the case of a3 ≡ 1.Namely wellposedness in L2 based Sobolev spaces dependson the balance of the regularizing dispersive effect froma3∂
3x and the destabilizing backward heat term a2∂
2x. This
balance was investigated in [T.A., A sharp condition forthe well-posedness of the linear KdV-type equation, (2012)].However, when the dispersive coefficient a3 degenerates theequation can be illposed without or despite the heat term,e.g. [Craig and Goodman, Linear dispersive equations ofAiry type,(1990)]. The techniques that applied in the non-degenerate case can be fruitfully applied in the degeneratesetting. In particular I will discuss the modified energymethod for the wellposedness arguments and the geomet-rical optics constructions for illposedness.
Timur AkhunovUniversity of [email protected]
MS19
Existence and Symmetry of Ground States to a
Boussinesq Abcd System
Abstract not available at time of publication.
Ming ChenUniversity of [email protected]
MS19
Coherent Frequency Dynamics for the PeriodicNonlinear Schrodinger Equation
We consider the cubic nonlinear Schrodinger equation on aperiodic box. By taking an appropriate large box limit ofthe equation (in a spirit similar to wave turbulence theory),we derive a (new) continuum equation on R
2, which turnsout to enjoy rather surprising symmetries, stability prop-erties, and even explicit solutions. This equation is thenused (via a rigorous approximation theorem) to better un-derstand the frequency dynamics of the original cubic NLSequation.
Zaher [email protected]
Pierre GermainNew York [email protected]
Erwan [email protected]
MS19
Approximation of Polyatomic FPU Lattices byKdV Equations
Famously, the Korteweg-de Vries equation serves as amodel for the propagation of long waves in Fermi-Pasta-Ulam (FPU) lattices. If one allows the material coeffi-cients in the FPU lattice to vary periodically, the “clas-sical” derivation and justification of the KdV equation goawry. By borrowing ideas from homogenization theory, wecan derive and justify an appropriate KdV limit for thisproblem. This work is joint with Shari Moskow, JeremyGaison and Qimin Zhang.
J. Douglas WrightDepartment of MathematicsDrexel [email protected]
MS20
On a Stochastic Leray-Alpha Model of Euler Equa-tions
We deal with the 3D inviscid Leray-alpha model. The wellposedness for this deterministic problem is not known; byadding a random perturbation we prove that there exists aunique (in law) global solution. The random forcing termformally preserves conservation of energy. The result holdsfor initial velocity of finite energy and the solution has finiteenergy almost surely. These results are easily extended tothe 2D case.
Hakima BessaihUniversity of Wyoming
PD13 Abstracts 83
MS20
Asymptotic Algebraic Spiral Solutions to the 2DIncompressible Euler Equations
Self similar solutions to the 2d incompressible Euler equa-tions can exhibit spiral behavior. Classical works ofKaden, Pullin, and Moore sought out asymptotic expan-sions through various methods and provided conjecturesfor the exponents of the leading terms in these expansions.We will prove that certain classes of self-similar solutionshave these asymptotic expansions and hence streamlinesexhibit spiral behavior in self-similar coordinates. This isjoint work with Volker Elling.
Michael Dabkowski, Volker EllingUniversity of [email protected], [email protected]
MS20
Well-Posedness of Euler-Like Evolution Equationsin Nonsmooth Domains and Relations to the Verti-cal Normal Modes Expansion of the Inviscid Prim-itive Equations
In previous joint work with C. Bardos and R. Temam,we show well-posedness of the Euler equations on a two-dimensional domain with acute corners, for weak solutionswith bounded initial vorticity. This problem is an approx-imation, near the zero flow, to the barotropic mode ofthe inviscid primitive equations, that is, the zeroth modeof the vertical normal modes expansion. As a furtherstep towards the understanding of the coupled barotropic-baroclinic system, we study the analogous approximatedproblem for more general background flows nearing theBrunt-Vaisala regime. The resulting system still resem-bles the two-dimensional Euler equation, though with non-standard boundary conditions, calling for further advance-ment of the elliptic theory tools developed in previouswork.
Francesco Di PlinioUniversity of Rome Tor [email protected]
Roger M. TemamInst. for Scientific Comput. and Appl. Math.Indiana [email protected]
MS20
Ill-Posedness for a Class of Equations Arising inHydrodynamics
We prove a linear ill-posedness result in L∞ for a transportequation with a non-local (singular integral) source term.Because we only use minimal assumptions on the velocityfield, we are able to prove non-linear ill-posedness resultsfor a large class of equations arising in hydrodynamics.
Tarek M. ElgindiNew York UniversityCourant Institute
MS21
An Asymptotic-Preserving Scheme for the Semi-conductor Boltzmann Equation toward the Energy-Transport Limit
We design an asymptotic-preserving scheme for the semi-conductor Boltzmann equation which leads to an energy-transport system for electron mass and internal energy asmean free path goes to zero. To overcome the stiffnessinduced by the convection terms, we adopt an even-odddecomposition to formulate the equation into a diffusiverelaxation system. New difficulty arises in the two-scalestiff collision terms, whereas the simple BGK penalizationdoes not work well to drive the solution to the correct limit.We propose a clever variant of it by introducing a thresholdon the stiffer collision term such that the evolution of thesolution resembles a Hilbert expansion at the continuouslevel. Formal asymptotic analysis and numerical resultsare presented to illustrate the efficiency and accuracy ofthe new scheme.
Jingwei HuThe University of Texas at [email protected]
Li WangUniversity of California, Los [email protected]
MS21
Asymptotic Preserving Scheme Based on Hyper-bolic Decomposition for Compressible Euler Equa-tions
We propose an asymptotic preserving, that is, stable andsufficiently accurate for all Mach number, central-upwindscheme for the compressible Euler equations of gas dynam-ics. It is well-known that the main difficulty associatedwith numerical simulations of such models is amplificationof the numerical viscosity for small Mach numbers, whichleads to a severe restriction on the size of time steps andthus substantially affects the efficiency of the method. Wepropose to decompose the underlying system of equationsinto two well-posed hyperbolic systems to remove the stiff-ness and thus to reduce the numerical viscosity and in-crease the size of time steps. The proposed discretizationprovides a consistent approximation of both the hyper-bolic compressible regime and the elliptic incompressibleregime. We conduct a number of one- and two-dimensionalnumerical experiments that demonstrate the asymptotic-preserving ”all-speed” properties of the proposed central-upwind scheme.
Alina ChertockNorth Carolina State UniversityDepartment of [email protected]
Shi JinShanghai Jiao Tong University, China and theUniversity of [email protected]
Alexander KurganovTulane UniversityDepartment of [email protected]
84 PD13 Abstracts
Jian-Guo LiuDuke [email protected]
MS21
Asymptotic Preserving Imex-Type Finite VolumeMethods for Some Geophysical Flows
We present new large time step methods for some hyperbolibalance laws. In particular we consider the Euler equa-tions in low Mach number limit and the shallow waterflows in the low Froude number limit. In order to takeinto account multiscale phenomena that typically appearin geophysical flows nonlinear fluxes are split into a linearpart governing the acoustic or gravitational waves and thenonlinear flow advection. We propose to approximate fastlinear waves implicitly in time and in space by means ofa genuinely multidimensional evolution operator. On theother hand, the nonlinear advection can be approximatedexplicitly in time and in space by means of the methodof characteristics or some standard numerical flux func-tion. Time integration is realized by the implicit-explicit(IMEX) method. We apply the IMEX Euler scheme, twostep Runge Kutta Cranck Nicolson scheme, as well asthe semi-implicit BDF scheme and prove their asymptoticpreserving property in the low Mach/Froude number lim-its. Numerical experiments demonstrate stability, accuracyand well-balanced behaviour of these new large time stepfinite volume schemes with respect to small Mach/Froudenumbers. This work has been done in cooperation withS. Noelle, K.R. Arun, L. Yelash and G. Bispen. Finan-cial support of the German Science Foundation under LU1470/2-2 is gratefully acknowledged.
Maria LukacovaUniversity of MainzInstitute of [email protected]
MS21
Uniform Stability and Well-balancing for Asymp-totic Preserving Schemes
For a class of the splittings we formally derive asymptoticconsistency and stability for IMEX discretizations of theEuler and shallow water equations. We also prove thatan elliptic equation for the water surface keeps lake-at-restsolutions well-balanced.
Sebastian NoelleIGPM - Institute for Geometry and PracticalMathematicsRWTH - Aachen University of [email protected]
Jochen SchuetzInstitute for Geometry and Practical MathematicsRWTH - Aachen University of [email protected]
MS22
Existence and Qualitative Properties of GroundsStates to the Non-Local Choquard-Type Equations
The Choquard equation, also known as the Hartree equa-tion or nonlinear Schrodinger-Newton equation is a station-ary nonlinear Schrodinger type equation where the nonlin-earity is coupled with a nonlocal convolution term given
by an attractive gravitational potential. We present sharpLiouville-type theorems on nonexistence of positive super-solutions of such equations in exterior domains. We alsodiscuss existence, positivity, symmetry and optimal decayproperties of ground state solutions under various assump-tions on the decay of the external potential and the shapeof the nonlinearity. In particular, we obtain a sharp decayestimate of the ground state solution which was discoveredby E.Lieb in 1977. This is a joint work with Jean VanSchaftingen (Louvain-la-Neuve, Belgium)
Vitaly MorozDepartment of MathematicsSwansea [email protected]
MS22
Dynamics of Fronts in Multistable Reaction-diffusion Equations
We study the dynamics of fronts for some reaction-diffusionequations having multi-stable reaction term. A typical ex-ample of such equations is the overdamped sine-Gordonequation. We will discuss generation of multiple fronts(with stairs-like shape), generation of metastable frontsand slow motion of fronts. This is a joint work with ToshikoOgiwara (Josai University)
Ken-Ichi NakamuraFaculty of Mathematics and PhysicsKanazawa [email protected]
MS22
A Local Regularity Theorem for Varifold MeanCurvature Flow
A family of k-dimensional surfaces in the n-dimensionalEuclidean space or Riemannian manifold (0 < k < n) iscalled the mean curvature flow (MCF) if the velocity ofmotion is equal to its mean curvature at each point andtime. While most research works on the MCF are done insmooth setting, one can define and study a weak notion ofMCF in the setting of varifold, known as Brakke’s MCF.I will explain its definition with relevant existence results,and then explain the partial regularity theorem.
Yoshihiro TonegawaHokkaido [email protected]
MS22
A Reaction-Diffusion Approach to the Existence ofGravity Water Waves
While variational techniques have been successfully appliedto capillary gravity water waves in order to prove existence,there are no corresponding existence proofs working in thephysical/original variables in the case of zero surface ten-sion. In this talk we will present a singular perturbationapproach to the existence of two-dimensional gravity waterwaves, which uses the original variables.
Georg WeissMathematical InstituteHeinrich Heine University
PD13 Abstracts 85
MS23
Focusing Quantum Many-body Dynamics: TheRigorous Derivation of the 1D Focusing Cubic Non-linear Schrdinger Equation
We consider the dynamics of N bosons in one dimension.We derive rigorously the one dimensional focusing cubicNLS with a quadratic trap as the N tends to infinity limitof the N-body dynamic and hence justify the mean-fieldlimit and prove the propagation of chaos for the focusingquantum many-body system.
Xuwen ChenBrown [email protected]
Justin HolmerDepartment of Mathematics, Brown University,Box 1917, 151 Thayer Street, Providence, RI 02912, [email protected]
MS23
Local Well-posedness of KdV with Potential
We prove local well-posedness of the Korteweg-de Vries(KdV) equation on the line with potential
∂tu = ∂x(−∂2xu− 3u2 + V u)
in H1(R). This is achieved by introducing a suitableparametrix approximation to the linear propagator, andthen proving local smoothing and maximal function esti-mates for this parametrix analogous to those obtained inthe free case by [C.E. Kenig, G. Ponce, and L. Vega, Well-posedness of the initial value problem for the Korteweg-deVries equation, J. Amer. Math. Soc. 4 (1991), no. 2, pp.323–347.] Global well-posedness follows easily from the en-ergy. This equation arises as a model of shallow water wavepropagation in a channel over a bottom of varying depth,and the H1(R) well-posedness is needed in the study ofsoliton dynamics.
Justin HolmerDepartment of Mathematics, Brown University,Box 1917, 151 Thayer Street, Providence, RI 02912, [email protected]
Mamikon GulianBrown Universitymamikon [email protected]
MS23
Scattering for Nonlinear Schrodinger Equationwith a Potential
We study global well-posedness and scattering for a 3d cu-bic focusing nonlinear Schrodinger equation
iut +Δu− V u+ |u|2u = 0; u(0) = u0 ∈ H1,
where V is a real-valued short-range potential with a smallnegative part. First, we find criteria for global well-posedness from the maximization problem related to theGagliardo-Nirenberg inequality. Next, we investigate scat-tering of such global solutions. Following Kenig-Merle’sprogram, we prove that (1) if there are non-scattering so-lutions obeying the criteria, then there exists a special
counterexample, namely a minimal blow-up solution; (2) ifwe further assume that a potential is repulsive, this coun-terexample does not exist, and the criteria for global well-posedness thus imply scattering.
Younghun HongUniversity of Texas at [email protected]
MS23
The Gross-Pitaevskii Hierarchy on the Three-dimensional Torus
In this talk, we will study the Gross-Pitaevskii hierarchyon the spatial domain T
3. In the first part of the talk, wewill prove a conditional uniqueness result for the hierarchy.As a result of our analysis, we will obtain a sharp rangeof integrability exponents in the key spacetime estimate.In the second part of the talk, we will add randomnessinto the problem by randomizing the collision operators onthe Fourier domain. In this randomized setting, we willstudy the limiting behavior of Duhamel iteration terms.The first part is based on joint work with Philip Gressmanand Gigliola Staffilani. The second part is based on jointwork with Gigliola Staffilani.
Vedran Sohinger, Philip GressmanUniversity of [email protected], [email protected]
Gigliola [email protected]
MS24
Geometric Partial Differential Equations in Ran-dom Media
We present some results on the geometry and regularityof some geometric variational problems in random media.As for the stationary problem, we study area minimiz-ers with random boundary condition, with focus on thePlateau Problem. For the evolutionary case, we considera mean curvature evolution in random media and relatedproblems that model population growth and phase transi-tions in random media.
Betul Orcan-EkmekciG.C. Evans [email protected]
MS24
Stochastic Homogenization for Porous MediumType Equations
In a recent paper, Ambrosio, Frid, and Silva study thehomogenization problem for a class of porous-medium typeequations where the flux function is a stationary randomprocess on a compact probability space. We extend theirresult to the general stochastic framework, removing thecompactness assumption on the probability space. Passingfrom the small to the large scale, we show the convergencein probability of the weak solutions of rescaled problem tothe weak solution of an homogenized deterministic porous-medium type equation.
Stefania PatriziUniversity of RomeRome, Italy
86 PD13 Abstracts
MS24
Boundary Regularity for a Class of DegenerateMonge-Ampere Equations
We discuss boundary regularity for the Monge-Ampereequation when the right hand side behaves as a positivepower of the distance function near the boundary of thedomain. As a consequence we obtain global C2 estimatesof the eigenfunction for the Monge-Ampere equation. Sim-ilar results were obtained in 2D by J.X. Hong, G. Huang,W. Wang.
Ovidiu SavinColumbia [email protected]
MS24
A New PDE Approach for Large Time Behavior ofHamilton-Jacobi Equations
I will present a new PDE approach to obtain large timebehavior of Hamilton- Jacobi equations. This applies tousual Hamilton-Jacobi equations, as well as the degenerateviscous cases, and weakly coupled systems. The degenerateviscous case was an open problem in last 15 years.
Hung TranUniversity of [email protected]
Filippo CagnettiDepartment of MathematicsInstituto Superior Tecnico, Lisbon, [email protected]
Diego GomesInstituto Superior Tecnico, Lisbon, [email protected]
Hiroyoshi MitakeFukuoka UniversityFukuoka, [email protected]
MS25
Uniform Stability of a Fluid-Structure Interactionwith Interface and Interior Controls in the FluidInside Structure Setting
We consider the uniform stability of finite energy solu-tions to a nonlinear fluid-structure interaction model in abounded domain Ω ∈ R2. In this model, Ω is occupied bytwo environments: fluid and solid, with fluid flown insidethe solid. The interaction occurs at the static interface.If an interior viscous nonlinear damping is placed on thesolid, the energy of the overall system decays to zero ata uniform rate. If additional boundary damping is placedon the interface and a geometric condition is satisfied, theenergy decays at an exponential rate.
Yongjin LuVirginia State [email protected]
Irena M. LasieckaUniversity of Virginia
MS25
Nonlinear Stability for Fluid-Structure InteractionModels
The classical one-phase Stefan problem models the tem-perature distribution in a homogeneous medium undergo-ing a phase transition, such as ice melting to water. Thisis accomplished by solving the heat equation on a time-dependent domain whose free-boundary is transported bythe normal derivative of the temperature along the evolv-ing and a priori unknown free-boundary. We establish aglobal-in-time stability result for small temperatures, usinga novel hybrid methodology, which combines energy esti-mates inspired by the Euler equations, decay estimates,and Hopf-type inequalities. This is joint work with M.Hadzic.
Steve ShkollerUniviversity of California, DavisDepartment of [email protected]
Mahir HadzicMassachusetts Institute of [email protected]
MS25
Shape Optimization with Nonlinear PDE Con-straints. Compressible Navier-Stokes Equations
Abstrat not available at time of publication.
Jan SokolowskiInstitut Elie CartanUniversite Henri Poincare Nancy [email protected]
MS25
Stability Analysis of Fluid-structure InteractionModels
We consider an established fluid-structure interactionmodel. We present recent joint results on polyno-mial/rational decay which are obtained as an interplay offunctional analytic techniques, PDE estimates, and micro-local analysis, to obtain the optimal parameter of decay.
Roberto TriggianiUniversity of [email protected]
MS26
Vortex Sheets in Domains with Boundaries
This talk concerns the interaction of incompressible 2Dflows with compact material boundaries under vortex sheetregularity. We focus on the dynamic behavior of the cir-culation of velocity around boundary components and thepossible exchange between flow vorticity and boundary cir-culation in flows with vortex sheet initial data. We re-cast 2D Euler evolution using vorticity and the circula-tion (around boundary components) as dynamic variablesand provide a weak formulation. Our main results are theequivalence between the weak velocity and weak vorticityformulations of the 2D Euler equations on domains with
PD13 Abstracts 87
boundary and a criterion for computing the resulting forceon boundary components for flows with little regularity.
Dragos IftimieUniversite de Lyon [email protected]
Milton Lopes Filho, Helena J. Nussenzveig LopesUniversidade Federal do Rio de [email protected], [email protected]
Franck SueurLaboratoire Jacques-Louis LionsUniversite Pierre et Marie [email protected]
MS26
Mathematical Models of Intermittency in Fully De-veloped Turbulence
Physical models of intermittency in fully developed turbu-lence use many phenomenological concepts such as activevolume, region, eddy, energy accumulation set, etc, usedto de- scribe non-uniformity of the energy cascade. In thistalk we give these terms a precise mathematical meaningin the language of the Littlewood-Paley analysis. With ourdefinitions we establish some of the deterministic counter-parts of the classical scaling laws for the energy spectrumand second order structure function with proper intermit-tency correction.
Roman ShvydkoyUniversity of Illinois at [email protected]
MS26
Inviscid Limit for Invariant Measures of the 2DStochastic Navier-Stokes Equations
We discuss recent results on the behavior in the infiniteReynolds number limit of invariant measures for the 2Dstochastic Navier-Stokes equations. We prove that the lim-iting measures are supported on bounded vorticity solu-tions of the 2D Euler equations. Invariant measures pro-vide a canonical object which can be used to link the fluidsequations to the heuristic statistical theories of turbulentflow.
Nathan Glatt-HoltzVirginia [email protected]
Vladmir SverakUniversity of [email protected]
Vlad C. VicolDept. Math - Princeton [email protected]
MS26
Non-Uniqueness Phenomena for Weak Solutions ofthe Euler Equations
It has been known since V. Scheffer’s groundbreaking worktwenty years ago that the Cauchy problem for the incom-pressible Euler equations admits non-unique weak solutionswith highly pathological behaviour. I will survey various
recent results on such ”wild solutions” and discuss conceiv-able uniqueness criteria.
Emil WiedemannDept. Math - University of British [email protected]
MS27
Stationary Co-Dimension One Structures in theFunctionalized Cahn-Hilliard Model
The functionalized Cahn-Hilliard energy models interfacialenergy in amphiphilic phase separated mixtures. Its mini-mizers encompass a rich class of morphologies with detailedinner structure, including bilayers, pore networks, pearledpores, and micelles. We address the existence and stabilityof a class of single-curvature bilayer structures in d > 1space-dimensions. The existence problem involves the con-struction of homoclinic solutions in a perturbed 4th-orderHamiltonian system. The linear stability involves an anal-ysis of meander and pearling modes.
Arjen DoelmanMathematisch [email protected]
Keith PromislowMichigan State [email protected]
Brian R. WettonUniversity of British ColumbiaDepartment of [email protected]
MS27
Existence and Stability of Solutions in the Multi-dimensional Swift–Hohenberg Equation
The existence of stationary localized spots for the pla-nar and the three-dimensional Swift–Hohenberg equationis proved using geometric blow-up techniques. The spotshave a much larger amplitude than that expected from aformal scaling in the far field. One advantage of the geo-metric blow-up methods is that the anticipated amplitudescaling does not enter as an assumption into the analysisbut emerges during the construction. The stability of thesesolutions will also be addressed.
Scott McCallaUniversity of California Los [email protected]
Bjorn SandstedeBrown Universitybjorn [email protected]
MS27
Pattern Selection in the Wake of Fronts
Motivated by the formation of complex patterns in thewake of growth processes in bacterial colonies, we’ll studymore generally patterns formed in the wake of movingfronts. We discuss a number of mathematical problemsthat arise when one tries to predict which wavenumbersand what type of pattern is created by the growth process.We therefore describe in some detail how to derive linearpredictions and analyze nonlinear mechanisms that lead to
88 PD13 Abstracts
failure of these predictions. This is joint work with MattHolzer and Ryan Goh.
Arnd ScheelUniversity of MinnesotaSchool of [email protected]
MS27
Small-amplitude Grain Boundaries of ArbitraryAngle in the Swift-Hohenberg Equation
We study grain boundaries in the Swift-Hohenberg equa-tion. Grain boundaries arise as stationary interfaces be-tween roll solutions of different orientations. Our analysisshows that such stationary interfaces exist near onset ofinstability for arbitrary angle between the roll solutions.
Qiliang WuUniversity of [email protected]
MS28
Global Existence for Water Wave Equations
Abstract not available at time of publication.
Pierre GermainNew York [email protected]
MS28
A Boundary Perturbation Method for Reconstruc-tion of Layered Media via Constrained QuadraticOptimization
The scattering of linear waves by layered media constitutea fundamental model in oceanography, acoustics, electro-magnetics, and the geosciences. In this talk we exam-ine the problem of detecting the geometric properties ofa two-dimensional acoustic medium where the fields aregoverned by the Helmholtz equation. Building upon thesuccess of our previous Boundary Perturbation approach(implemented with the Operator Expansions formalism)we derive a new approach which augments this with a new”smoothing” mechanism. With numerous numerical exper-iments we demonstrate the enhanced stability and accuracyof our new approach which further suggests not only a rig-orous proof of convergence, but also a path to generalizingthe algorithm to multiple layers, three dimensions, and thefull equations of linear elasticity and Maxwell’s equations.
David P. NichollsUniversity of Illinois at [email protected]
Alison [email protected]
MS28
Global Existence for Gravity Waves in 2 Dimension
We consider the water waves system for the evolution ofa perfect fluid with a free surface in 2d, under the influ-ence of gravity. For sufficiently smooth and localized initialdata we prove global existence of small solutions. We also
prove asymptotics for such solutions, showing a nonlinearbehavior as time goes to infinity. This is joint work withA. Ionescu.
Fabio PusateriPrinceton [email protected]
MS28
On the Finite-time Splash and Splat Singularitiesfor the 3-D Free-surface Euler Equations
e prove that the 3-D free-surface incompressible Eulerequations with regular initial geometries and velocity fieldshave solutions which can form a finite-time splash (or splat)singularity, wherein the evolving 2-D surface, the movingboundary of the fluid domain, self-intersects at a point (oron surface). Such singularities can occur when the crest ofa breaking wave falls unto its trough, or in the study of dropimpact upon liquid surfaces. Our approach is founded uponthe Lagrangian description of the free-boundary problem,combined with a novel approximation scheme of a finitecollection of local coordinate charts; as such we are able toanalyze a rather general set of geometries for the evolving2-D free-surface of the fluid.
Steve ShkollerUniviversity of California, DavisDepartment of [email protected]
Daniel CoutandHeriot-Watt [email protected]
MS29
On the Bound of DG and Central DG Operators
Discontinuous Galerkin (DG) and central DG methods aretwo families of high order methods, and it was observedthat the latter allows larger time steps in stable numericalsimulation especially when the accuracy is relatively high.In this paper, we estimate the bounds of DG and centralDG spatial operators for linear advection equation. Basedon such estimates and Kreiss-Wu theory, time step condi-tions are obtained to ensure numerical stability when themethods are coupled with locally stable time discretiza-tions. In particular, with a fixed time discretization, thetime step grows quadratically with k when the DG methodis used in space (here k is the polynomial degree of the dis-crete space), while in the case of the central DG method,the dependence of the time step on k is linear. In addition,the analysis provides new insight into the role of a param-eter in the central DG formulation. We verify our resultsnumerically, and also discuss the extension of the analysisto some related discretiztions.
Fengyan LiRensselaer Polytechnic [email protected]
Matthew ReynaRensselaer Polytechnic InstituteDepartment of Mathematical [email protected]
MS29
High-order Multiderivative Time Integrators for
PD13 Abstracts 89
Hyperbolic Conservation Laws
Time stepping methods for hyperbolic conservation lawsoften fall into one of two distinct categories: a method oflines formulation, and a single-step Taylor (Lax-Wendroff)formulation. The existence of so-called multiderivativetime integrators for ordinary differnetial equations indicatethat this is indeed a false dichotomy. In fact, multideriva-tive methods have a long history of development for or-dinary differential equations dating back to at least the1950’s. However, only the extreme versions of these meth-ods have been explored as a numerical tool for solving par-tial differential equations. In this work, we demonstratethat explicit Runge-Kutta methods as well as high-orderTaylor (Lax-Wendroff) methods are special cases of multi-derivative methods, and in addition, we demonstrate howthese methods can be applied for solving hyperbolic conser-vation laws using the weighted essentially non-oscillatory(WENO) method, as well as the discontinuous Galerkin(DG) method.
David C. Seal, Yaman GucluDepartment of MathematicsMichigan State [email protected], [email protected]
Andrew J. ChristliebMichigan State UniverityDepartment of [email protected]
MS29
A High Order Weno Scheme for Detonation Waves
Abstract not available at time of publication.
Wei WangFlorida International University, Miami, [email protected]
MS29
Asymptotic Preserving Discontinuous GalerkinMethod for the Radiative Transfer Equation
Many kinetic equations converge to macroscopic models,known as the asymptotic limit of the kinetic equations,when ε (the ratio of mean free path over macroscopicsize) − > 0. Asymptotic preserving (AP) methods aredesigned to preserve the asymptotic limits from micro-scopic to macroscopic models in the discrete setting. Inthe asymptotic regimes (when ε <<0) , they become a con-sistent discretization of the macroscopic solvers automati-cally, while remain microscopic otherwise. In this presen-tation, we consider the radiative transfer equation whichmodels isotropic particle scattering in a medium. Manyfinite volume (FV) methods do not have the AP prop-erty. Discontinuous Galerkin (DG) (with upwind flux andat least piecewise linear polynomial) methods are knownto have such property, but require much more degree offreedom, especially in high dimensional applications. Wewill present an AP mixed DG-FV method which has com-parable computational cost and memory as FV method.Rigorous analysis will be provided to show that the pro-posed methods are consistent with the limit equation inthe asymptotic regimes. Some numerical examples are pre-sented to demonstrate the performance of our methods.
Yulong XingDepartment of Mathematics
Univeristy of Tennessee / Oak Ridge National [email protected]
MS30
Conditioning of Nonlocal Operators in FractionalSobolev Spaces
We study the conditioning of the weak from of the non-local operator associated to nonlocal diffusion and peri-dynamics in fractional Sobolev spaces. For both maximaland minimal eigenvalues, we provide sharp estimates whichinvolve explicit quantifications of the horizon, mesh size,and the regularity of the fractional Sobolev space. Forthese estimates, we use an interesting combination of toolsfrom functional analysis and numerical linear algebra. Thesharpness results are also supported with numerical exper-iments.
Burak AksoyluTOBB University of Economics and Technology, Dept ofMathLouisiana State University, Dept of [email protected]
Zuhal UnluLouisiana State [email protected]
MS30
Adaptive Numerical Methods for Periydnamics
Abstract not available at time of publication.
Qiang DuPenn State UniversityDepartment of [email protected]
MS30
Differentiability for Solutions of Nonlocal Modelswith Weakly Singular Kernels
Abstract not available at time of publication.
Mikil FossUniversity of Nebraska-LincolnLincoln, [email protected]
MS30
Surface Effects and Affects on Ordinary IsotropicPeridynamics Models
Abstract not available at time of publication.
John A. MitchellSandia National [email protected]
MS31
Acoustic Propagation in a Saturated Piezo-Elastic,Porous Medium
We study the problem of derivation of an effective modelof acoustic wave propagation in a two-phase, non-periodicmedium modeling a fine mixture of linear piezo-elastic solid
90 PD13 Abstracts
and a viscous Newtonian, ionic bearing fluid. Bone tis-sue is an important example of a composite material thatcan be modeled in this fashion. We develop two-scale ho-mogenization methods for this system, and discuss also astationary random, scale-separated microstructure. Theratio e of the macroscopic length scale and a typical sizeof the microstructural inhomogeneity is a small parameterof the problem. Another possibly small parameter is thePeclet number which in?uences the type of e?ective equa-tions which are obtained.
Robert P. GilbertDepartment of Mathematical SciencesUniversity of [email protected]
MS31
Homogenization of Rigid Suspensions with HighlyOscillatory Velocity-Dependent Surface Forces
We study particulate flows or suspensions of solid particlesin a viscous incompressible fluid at the presence of highlyoscillatory surface forces. The flow at a small Reynoldsnumber is modeled by the Stokes equations coupled withthe motion of absolutely rigid particles arranged in a pe-riodic array. The objective is to perform homogenizationfor the given suspension and obtain an equivalent descrip-tion of a homogeneous (effective) medium whose viscosity isdetermined using the analysis on a periodicity cell. Math-ematical tools that the main technical construct is builtupon are based on Γ-convergence theory.
Yuliya GorbDepartment of MathematicsUniversity of [email protected]
Florin [email protected]
Bogdan M. VernescuWorcester Polytechnic InstDept of Mathematical [email protected]
MS31
Nonlinear Neutral Inclusions: Assemblages ofSpheres and Confocal Coated Ellipsoids
If a neutral inclusion is inserted in a matrix containinga uniform applied electric field, it does not disturb thefield outside the inclusion. The well known Hashin coatedsphere is an example of a neutral coated inclusion. In thistalk, we consider the problem of constructing neutral in-clusions from nonlinear materials. In particular, we discussassemblages of coated spheres and ellipsoids.
Silvia Jimenez BolanosDepartment of Mathematical SciencesWorcester Polytechnic [email protected]
Bogdan M. VernescuWorcester Polytechnic InstDept of Mathematical Sciences
MS31
Dynamic Brittle Fracture as a Small Horizon Limitof Peridynamics
We consider peridynamic formulations with unstable con-stitutive laws that soften beyond a critical stretch. Herewe discover new quantitative and qualitative informationthat is extracted from the peridynamic formulation us-ing scaling arguments and by passing to a distinguishedsmall horizon limit. In this limit the dynamics correspondto the simultaneous evolution of elastic displacement andfracture. The displacement fields are shown to satisfy thewave equation. The wave equation provides the dynamiccoupling between elastic waves and the evolving fracturepath inside the media. The limit evolutions have boundedenergy expressed in terms of the bulk and surface energiesof brittle fracture mechanics.
Robert P. LiptonDepartment of MathematicsLouisiana State [email protected]
MS32
On the Well-Posedness of the 3D Navier-StokesEquations in the Largest Critical Space
In this talk I will discuss various well-posedness and ill-posedness results for the 3D Navier-Stokes equations in thelargest critical space. In particular, I will describe a recentnorm inflation result for the hyperdissipative Navier-Stokesequations that occurs in critical and supercritical spaceseven in the case where the global regularity is known. Wewill see how a new non-critical scaling becomes an obstaclein proving small initial data results in critical spaces.
Alexey CheskidovUniversity of Illinois [email protected]
MS32
Initial-Boundary Layer in the Nonlinear Darcy-Brinkman System
We study the interaction of initial layer and boundary layerin the nonlinear Darcy-Brinkman system in the vanishingDarcy number limit. In particular, we show the existenceof a function of corner layer type (so called initial-boundarylayer) in the solution of the nonlinear Darcy-Brinkmansystem. An approximate solution is constructed by themethod of asymptotic expansion. We establish the opti-mal convergence rates in various Sobolev norms via energymethod.
Daozhi HanFlorida State [email protected]
Xiaoming [email protected]
MS32
Gevrey Regularity of the Critical and Supercritical
PD13 Abstracts 91
Quasi-Geostrophic Equations
The dissipative quasi-geostrophic (QG) equation is an im-portant model in geophysical fluid dynamics, but alsomathematically important since it can be considered as a2D model of the 3D incompressible Navier-Stokes equa-tions. Consequently, it has received much attention inrecent years. Studies of this equation are usually sepa-rated into three cases: supercritical, critical, and subcrit-ical, where in each case the order of dissipation increases.By now, the subcritical case is well understood; global ex-istence of weak solutions was established in [Resnick, ’95],while global existence of a unique regular solution with ini-tial data in Lp-spaces, for instance, was established in [Wu,’01]. Recently, global well-posedness for arbitrary smoothinitial data was established by several authors indepen-dently: [Caffarelli-Vasseur, ’06], [Kiselev-Nazarov-Volberg,’07], and [Constantin-Vicol, ’11]. However, there is stillmuch that is not known in the supercritical and even, crit-ical case. Our paper explores higher-order regularity of thesupercritical and critical QG equation. In particular, weestablish Gevrey regularity of solutions in Lp-based Besovspaces. The techniques we use are based on the ideas of[Miura, ’06] and an extension of those in [Biswas, preprint],where the main idea is to establish control on the nonlinearterm via a suitable commutator estimate. Since we seek Lp
bounds, we adapt the approach of [Lemarie-Rieusset, ’99]and view the commutator as a bilinear multiplier operator.However, to establish these bounds, we must also adopt thetechniques of Michael Lacey and Christoph Thiele used tothe bound the Bilinear Hilbert Transform. As an applica-tion, we also deduce higher-order decay estimates on thesolution.
Animikh BiswasUniversity of North Carolinaat [email protected]
Vincent Martinez, Prabath SilvaIndiana University, [email protected], [email protected]
MS33
High Order Finite Elements for Wave Propagation:Like It Or Lump It?
High order finite element methods have long been usedfor computational wave propagation for both first orderand second order equations alike. A key issue with com-putational wave propagation is the phase accuracy of themethods: getting the waves to propagate with the rightspeed is often at least as, if not more, important than theconvergence rate. The main variants are the finite elementmethod (FEM) and spectral element method (SEM) witheach technique having its band of advocates. SEM can beviewed as a higher order mass-lumped FEM. Despite thepredominance of these methods, there is comparatively lit-tle by way of hard analysis on what each variant offersin terms of phase accuracy, and there is considerable mis-information and confusion in the literature on this topic.In this presentation, we shall attempt to shed some lighton the matter, and also briefly mention new methods thatimprove on both FEM and SEM
Mark AinsworthDivision of Applied MathematicsBrown University
Mark [email protected]
MS33
An Element Conservative Dpg Method forConvection-Dominated Problems
Abstract not available at time of publication.
Truman E. EllisICESUniversity of Texas at [email protected]
MS33
Dissipative Methods for Wave Equations in SecondOrder Form
We consider the construction and analysis of high-orderdissipative elements for solving time-domain wave equa-tions in second order form. These include natural gener-alizations of dissipative methods for first order hyperbolicsystems including upwind discontinuous Galerkin methodsas well as Hermite methods.The essential ingredient in thegeneralization is to consider the standard energy form forsecond order wave equations as well as its space derivatives.We demonstrate that the proposed methods can be stablyhybridized.
Thomas M. HagstromSouthern Methodist UniversityDepartment of [email protected]
Daniel AppeloDepartment of Mathematics and StatisticsUniversity of New [email protected]
MS33
Mixed Finite Elements for Wave Equations
I will survey several of the issues facing numerical approx-imation of wave equations by finite element methods –before considering stability and accuracy, we must decidewhether to use first- or second-order forms of the equa-tions. We must deal with numerical dispersion, invert-ing mass matrices at each time step and energy conserva-tion/dissipation properties of our numerical solutions. Af-ter setting the stage for many of the presentations of theminisymposium, I will present some recent results alongthese lines for mixed finite element discretization of theacoustic wave and linear shallow water equations. Theseinclude nearly-conservative symplectic time-stepping, con-version between first- and second-order, and the effect of adrag term on energy balances.
Robert C. KirbyBaylor UniversityRobert [email protected]
MS34
Analysis of the Two-Phase Flow in Rotating Hele-Shaw Cells with Coriolis Effects
The free boundary problem of a two phase flow in a rotatingHele-Shaw cell with Coriolis effects is studied. Existenceand uniqueness of solutions near spheres is established, and
92 PD13 Abstracts
the asymptotic stability and instability of the trivial solu-tion is characterized in dependence on the fluid densities.
Patrick GuidottiUniversity of California at [email protected]
Joachim EscherLebiniz Universitat Hannover (Germany)[email protected]
Christoph WalkerLeibniz University [email protected]
MS34
Crystal Facets: From Microscale Motion toSingular-Diffusion Pdes
Continuum theories of epitaxial growth usually breakdown in the vicinity of macroscopically flat surface re-gions (facets). In surface diffusion, such facets give riseto free-boundary problems with non-trivial microstructurefor fourth-order singular-diffusion PDEs. In this talk, Iwill present recent progress in coupling facet motion withdiscrete (microscale) schemes for surface atomic defects(steps) under surface diffusion.
Dionisios MargetisUniversity of Maryland, College [email protected]
MS34
Examples of Singular Diffusion Equations in Oneand Two Dimension: Facets and More
We focus on a simple singular diffusion equation (i.e. atotal variation flow in 1-d) and its counterpart regularizedby additive linear diffusion. We study these two equationsin one and two dimensions. After quickly establishing ex-istence, uniqueness and regularity we concentrate on cre-ation, evolution and interaction of facets. We also exhibita number of explicit solutions.
Mucha PiotrWarsaw University Mathematics DepartmentMathematics [email protected]
Muszkieta MonikaWroclaw University of TechnologyInstitute of Mathematics and Computer [email protected]
Piotr RybkaWarsaw UniversityMathematics [email protected]
MS34
On Singular Perturbation Limit of the Allen-CahnEquation
We present some results on the characterization of energyconcentration measure for the Allen-Cahn equation as thethickness of interface approaches to zero. The limiting in-terface moves with velocity equals to its mean curvature
plus given ambient vector field with low regularity. This isa joint work with Keisuke Takasao.
Yoshihiro TonegawaHokkaido [email protected]
MS35
Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti - PartII
The derivation of nonlinear dispersive PDE, such as thenonlinear Schrodinger (NLS) or nonlinear Hartree (NLH)equations, from many body quantum dynamics is cen-tral topic in mathematical physics, which has been ap-proached by many authors in a variety of ways. In par-ticular, one way to derive NLS is via the Gross-Pitaevskii(GP) hierarchy, which is an infinite system of coupled lin-ear non-homogeneous PDE. The most involved part in sucha derivation of NLS consists in establishing uniqueness ofsolutions to the GP. That was achieved in seminal papers ofErdos-Schlein-Yau. Recently, with Hainzl and Seiringer weobtained a new, simpler proof of the unconditional unique-ness of solutions to the cubic Gross-Pitaevskii hierarchy inR
3. One of the main tools in our analysis is the quan-tum de Finetti theorem. In the first talk, Pavlovic willpresent a brief review of the derivation of NLS via the GP,describing the context in which the new uniqueness resultappears. In the second talk, Chen will illustrate how quan-tum de Finetti’s theorem can be used to obtain uncondi-tional uniqueness of the GP.
Thomas ChenUniversity of Texas at [email protected]
MS35
The 2d Schrodinger Equation on Irrational Tori
Abstract not available at time of publication.
Seckin DemirbasUniversity of [email protected]
MS35
Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti-PartI
The derivation of nonlinear dispersive PDE, such as thenonlinear Schrodinger (NLS) or nonlinear Hartree (NLH)equations, from many body quantum dynamics is cen-tral topic in mathematical physics, which has been ap-proached by many authors in a variety of ways. In par-ticular, one way to derive NLS is via the Gross-Pitaevskii(GP) hierarchy, which is an infinite system of coupled lin-ear non-homogeneous PDE. The most involved part in sucha derivation of NLS consists in establishing uniqueness ofsolutions to the GP. That was achieved in seminal papersof Erdos-Schlein-Yau. Recently, with Chen, Hainzl andSeiringer we obtained a new, simpler proof of the uncondi-tional uniqueness of solutions to the cubic Gross-Pitaevskiihierarchy in R
3. One of the main tools in our analysis isthe quantum de Finetti theorem. In the first talk, Pavlovicwill present a brief review of the derivation of NLS via GPdescribing the context in which the new uniqueness result
PD13 Abstracts 93
appears. In the second talk, Chen will illustrate how quan-tum de Finetti’s theorem can be used to obtain uncondi-tional uniqueness of the GP.
Natasa PavlovicUniversity of Texas at [email protected]
MS35
Blow Up Solutions in the Focusing NLS
Abstract not available at time of publication.
Svetlana RoudenkoThe George Washington [email protected]
MS36
Wide-Stencil and Filter Schemes for Monge Am-pere Equations
Abstract not available at time of publication.
Adam ObermanMcGill [email protected]
MS36
Optimal Transport with Proximal Splitting
We here present the use of first order convex optimizationschemes to solve the discretized dynamic optimal transportproblem, initially proposed by Benamou and Brenier. Wedevelop a staggered grid discretization that is well adaptedto the computation of the L2 optimal transport geodesicbetween distributions defined on a uniform spatial grid.We show how proximal splitting schemes can be used tosolve the resulting large scale convex optimization problem.A specific instantiation of this method on a centered gridcorresponds to the initial algorithm developed by Benamouand Brenier. We also show how more general cost functionscan be taken into account and how to extend the methodto perform optimal transport on a Riemannian manifold.
Nicolas PapadakisLJK [email protected]
MS36
Optimal Transportation with Infinitely ManyMarginal
Abstract not available at time of publication.
brendan PassU. [email protected]
MS36
Optimal Transport for Illumination Optics
The design of freeform reflectors and lenses for illuminationsystems has become very relevant with the rise of LEDs.The problem is closely related to optimal transport. Wedeveloped a method to design convex or concave reflec-tors based on a novel numerical method for the Monge-Ampere equation and a numerical Legendre-Fenchel trans-
form. I will give an example of a reflector that projects afamous Dutch painting on a wall from a uniform parallellight beam.
Corien PrinsPhilips Research [email protected]
Jan Ten Thije BoonkkampDepartment of Mathematics and Computer ScienceEindhoven University of [email protected]
Wilbert IJzermanPhilips [email protected]
Teus TukkerPhilips [email protected]
Jarno van RoosmalenEindhoven University of [email protected]
MS37
Exact Controllability of a Membrane Immersed ina Potential Fluid
We consider the problem of controlling a membrane orplate that is either immersed in a (linearized) potentialfluid,or forms a portion of the boundary of a potentialfluid. The control acts along a portion of the boundaryof the membrane. We show that for sufficiently small fluiddensity, and time large enough (related to the geometryof the membrane), exact controllability holds in the finiteenergy space.
Scott HansenIowa State UniversityDepartment of [email protected]
MS37
On Incompressible Two-Phase Fluid Flows withPhase Transitions
We consider a sharp interface model for two-phase flowswith surface tension undergoing phase transitions. Themodel is based on conservation of mass, momentum andenergy, and hence is physically exact. It further employsthe standard constitutive law of Newton for the stress ten-sor, Fourier’s law for heat conduction, and it is thermody-namically consistent. We establish well-posedness of themodel and study the qualitative behavior of solutions. Inparticular, we characterize all equilibria and study theirstability properties. The total entropy turns out to be animportant quantity in the stability analysis.
Gieri SimonettVanderbilt [email protected]
MS37
On the Two-phase Navier-Stokes Equations inCylindrical Domains
We consider the dynamics of two incompressible fluids in
94 PD13 Abstracts
a cylindrical domain, separated by a sharp interface. Thecontact angle is assumed to be constant and equal to 90degrees. We prove the well-posedness of the correspondingfree boundary problem and we present results concerningthe stability of equilibria (Rayleigh-Taylor instability). Fi-nally we present a result on bifurcation at multiple eigen-values.
Mathias WilkeUniversity of [email protected]
MS37
Min-Max Game Problem for Elastic and Visco-Elastic Fluid Structure Interactions
We present the features of a min-max game theory prob-lem for a linear fluid-structure interaction model in bothelastic and visco-elastic case, with (possibly) control anddisturbance exercised at the interfce between the two me-dia. The sought-after saddle solutions are expressed in apointwise feedback form, which involves a Riccati opera-tor; that is, an operator satisfying a suitable non-standardRiccati differential equation.
Jing ZhangVirginia State [email protected]
Irena M. Lasiecka, Roberto TriggianiUniversity of [email protected], [email protected]
MS38
On the Behavior of Bounded Vorticity, BoundedVelocity Solutions to the 2D Euler Equations
I will present recent work characterizing all possible weaksolutions to the 2D Euler equations in the full plane or theexterior of a single obstacle having bounded velocity andbounded vorticity. The class of all such solutions general-izes the solutions obtained originally by Phillipe Serfati in1995 for the full plane, which have sublinear growth of thepressure at infinity. For more general solutions a conditionat infinity, in terms of the velocity or the pressure, holdsweakly, and the circulation about the obstacle can vary foran exterior domain. These results build on those of jointwork with Ambrose, Lopes Filho, and Nussenzveig Lopes.
James P. KelliherUC [email protected]
MS38
On Vorticity Formulation for the Navier-StokesEquations in the Half Plane and Its Applicationsto the Inviscid Limit Problem
We consider the Navier-Stokes equations for viscous incom-pressible flows in the half plane under the no-slip boundarycondition. By using a vorticity formulation we prove time-local convergence of the Navier-Stokes flows to the Eulerflows outside a boundary layer and to the Prandtl flows inthe boundary layer at the inviscid limit when the initialvorticity is located away from the boundary.
Yasunori MaekawaDept. Math , Tohoku University
MS38
Planar Limits of 3D Helical Flows
We study the limits of three-dimensional helical viscousand inviscid incompressible flows in an infinite circularpipe, with respectively no-slip and no-penetration bound-ary conditions, as the step approaches infinity. We showthat, as the step becomes large, the three-dimensional he-lical flow approaches a planar flow, which is governed bythe so-called two-and-half Navier-Stokes and Euler equa-tions, respectively. The step or pitch is the magnitude ofthe translation after rotating one full turn around the sym-metry axis.
Anna MazzucatoDepartment of Mathematics, Penn State [email protected]
Milton C. Lopes FilhoUniversidade Federal do rio de [email protected]
Dongjuan NiuCapital Normal University, [email protected]
Helena J. Nussenzveig LopesUniversidade Federal do Rio de [email protected]
Edriss S. TitiUniversity of California, Irvine andWeizmann Institute of Science, [email protected], [email protected]
MS38
Remarks on the Question of Dissipation Anomalyfor the Navier-Stokes and Euler Equations
Dissipation Anomaly is one of the well-defined mathemati-cal problems in turbulence theory. It states that the meanrate of dissipation of energy of viscous turbulent flows re-mains bounded away from zero, as the viscosity tends tozero. We will provide simple examples demonstrating thisphenomenon, and explaining the idea behind the availablerigorous results. These examples illustrate that the currentmathematical formulation might not be adequate; and an-other formulation is proposed ruling out these examples.
Edriss S. TitiUniversity of California, Irvine andWeizmann Institute of Science, [email protected], [email protected]
Claude BardosLaboratory J.L. Lions, Universit{e} Pierre et Marie CurieParis, 75013, [email protected]
MS39
Title Not Available at Time of Publication
Abstract not available at time of publication.
Simon Arridge
PD13 Abstracts 95
Department of Computer ScienceUniversity College [email protected]
MS39
Deriving the Kubelka-Munk Equations from Ra-diative Transport
Kubelka-Munk theory provides a simple model to describefor light propagation in turbid media. We derive this the-ory systematically from the theory of radiative transporttheory by analyzing the system of equations resulting fromapplying the double spherical harmonics method. Fromthis systematic derivation, we establish theoretical basis ofthe Kubelka-Munk equations which has been an outstand-ing issue. Moreover, we extend this theory to study severalproblems of practical interest.
Arnold D. KimUniversity of California, [email protected]
Chris SandovalApplied MathematicsUniversity of California, [email protected]
MS39
Case’s Method in Three Dimensions
Case’s method obtains solutions to the radiative transportequation as a linear combination of elementary solutionsor singular eigenfunctions. In this talk, we will extendthe method to three dimensions with arbitrary anisotropicscattering. We evaluate singular eigenfunctions in refer-ence frames whose z-axes are taken in the direction of eachwave vector. By doing so, the three-dimensional equationreduces to the one-dimensional equation.
Manabu MachidaDepartment of MathematicsUniversity of [email protected]
MS39
Filtered Spherical Harmonics Expansions of theRadiative Transfer Equation
Recent work has shown that various filtering schemes cangreatly improve the robustness of moment-based methodsfor solving the radiative transfer equation. In this talkI will outline several approaches used to filter sphericalharmonics moments and apply these filters to the transportsolutions for problems of linear particle transport as wellhas nonlinear x-ray transport. I will also detail how filterscan be employed to improve discrete ordinates methodsand talk about nonlinear extensions.
Ryan G McClarrenTexas A & [email protected]
MS40
On the Benjamin-Feir Instability
I will speak on the Benjamin-Feir (or modulational) insta-bility of Stokes periodic waves on deep water. I will begin
by describing a variational framework that I recently de-veloped with J. Bronski and determine instability to longwavelengths perturbations for a general class of Hamil-tonian systems, allowing for nonlocal dispersion. I willexplain an asymptotic approach that M. Johnson and Iadapted for Whitham’s equation of surface water waves,qualitatively reproducing the Benjamin-Feir instability ofStokes waves. I will discuss on how to extend the resultsto the exact, water wave problem.
Vera Mikyoung HurUniversity of Illinois at [email protected]
Jared BronskiUniversity of Illinois Urbana-ChampaignDepartment of [email protected]
Mathew JohnsonUniversity of [email protected]
MS40
Pressure Beneath a Traveling Wave with ConstantVorticity
In this talk I will give a brief overview of two-dimensionaltraveling gravity waves with constant vorticity, with andwithout stagnation points. Following this, I will present adirect relationship between the surface profile of the travel-ing wave and the pressure at the bottom of the fluid. Thisrelationship is obtained as a direct consequence of the fullequations describing the traveling wave without approxi-mation. Using this relationship, we reconstruct both thepressure and streamlines throughout the fluid domain. Inparticular, we can recover local minima of the pressure (be-low the atmospheric value) beneath the crest in the pres-ence of a stagnation point.
Vishal VasanThe Pennsylvania State [email protected]
Katie OliverasSeattle [email protected]
MS40
Mathematical Theory of Wind-generated WaterWaves
It is easy to see that wind blowing over a body of water cancreate waves. But this simple observation leads to a morefundamental question: Under what conditions on the veloc-ity profile of the wind will persistent surface water waves begenerated? This has been problem has been studied inten-sively in the applied fluid dynamics community since thefirst efforts of Kelvin in 1871. In this talk, we will presenta mathematical treatment of the predominant model forwind-wave generation, the so-called quasi-laminar modelof J. Miles. Essentially, this entails determining the (lin-ear) stability properties of the family of laminar flow solu-tions to the two-phase interface Euler equation. We givea rigorous derivation of the linearized evolution equationsabout an arbitrary steady solution, and, using this, we givea complete proof of the celebrated instability criterion ofMiles. In particular, our analysis incorporates both the ef-fects of surface tension and a vortex sheet on the air–sea
96 PD13 Abstracts
interface. We are thus able to give a unified equation con-necting the Kelvin–Helmholtz and quasi-laminar models ofwave generation.
Samuel WalshNew York [email protected]
Oliver Buhler, Jalal ShatahCourant Institute of Mathematical SciencesNew York [email protected], [email protected]
Chongchun ZengGeorgia [email protected]
MS40
Large-Amplitude Solitary Waves Generated bySurface Pressure
We study solitary traveling waves in a two-dimensional in-viscid incompressible fluid bounded below by a horizontalbed and above by a free surface on which a non-constantpressure is applied. By varying the surface pressure, weconstruct large-amplitude waves of elevation and depres-sion. We assume that the wave and surface pressure aresymmetric, and that the Froude number of the underlyingflow is sufficiently large (supercritical).
Miles WheelerDepartment of MathematicsBrown [email protected]
MS41
On the Parametrized Maximum Principle FluxLimiters for Hyperbolic Conservation Laws: Ac-curacy and Application
Maximum principle preserving is an important property ofthe entropy solution, the physically relevant solution, tothe scalar hyperbolic conservation laws. One would liketo imitate the property on the numerical level. On theother hand, preserving the maximum principle also pro-vides a stability to the numerical schemes for solving theconservation law problem. In this talk, we will discusshow this problem is addressed by introducing a series ofmaximum principle constraint. By decoupling those con-straints, a parametrized flux limiting technique is devel-oped to make sure the numerical solution preserves maxi-mum principle in the conservative and consistent mannerwhile the scheme is still high order. Generalization of thetechnique to convection-diffusion problem will also be dis-cussed.
Zhengfu XuMichigan Technological UniversityDept. of Mathematical [email protected]
MS41
Stability Analysis and Error Estimates of an Ex-actly Divergence-Free Method for the Magnetic In-duction Equations
In this talk, we consider an exactly divergence-free schemeto solve magnetic induction equations. This problem is ex-
tracted from the numerical simulations of ideal Magneto-hydrodynamics (MHD) equations, which contain nonlinearhyperbolic equations as well as a divergence-free conditionfor the magnetic field. Numerical methods without satisfy-ing such condition may lead to numerical instability. Oneclass of methods called constrained transport schemes iswidely used as a divergence-free treatment. However, whythese schemes work is still not fully understood. In thiswork, we take the exactly divergence-free schemes proposedby Li and Xu as a candidate of the constrained transportschemes, and analyze the stability and errors when solvingmagnetic induction equations. This is the most significantpart to understand the divergence-free treatment in MHDsimulations. Our result can not only explain the stabilitymechanism of this particular scheme and the role of the ex-actly divergence-free condition in that, but also provide theinsight to understand other constrained transport schemes.
Fengyan LiRensselaer Polytechnic [email protected]
He YangRensselaer Polytechnic InstituteDepartment of Mathematical [email protected]
MS41
High Order Fast Sweeping and Homotopy Methodswith Linear Computational Complexity for SolvingSteady State Problems of Hyperbolic PDEs
In this talk, I present our recent studies on developing effi-cient high order numerical methods for solving steady stateproblems of two classes of hyperbolic PDEs: Eikonal equa-tions and hyperbolic conservation laws. The methods wepropose have linear computational complexity, namely, thecomputational cost is O(N) where N is the number of gridpoints of the computational mesh. For Eikonal equations,we design a third order fast sweeping method with linearcomputational complexity. This iterative method utilizesthe Gauss-Seidel iterations and alternating sweeping strat-egy to cover a family of characteristics of the Eikonal equa-tions in a certain direction simultaneously in each sweepingorder. The method is based on a third order discontinuousGalerkin (DG) finite element solver. Novel causality indi-cators are designed to guide the information flow directionsof the nonlinear Eikonal equations. Numerical experimentsshow that the method has third order accuracy and a lin-ear computational complexity. For hyperbolic conservationlaws, a homotopy continuation method is applied to solvepolynomial systems resulting from a third order weightedessentially non-oscillatory (WENO) discretization of thesystem. Via numerical experiments for both scalar and sys-tem problems in one and two dimensions, we show that thisnew approach has linear computational complexity and isfree of the CFL condition constraint.
Yongtao ZhangUniversity of Notre [email protected]
MS41
Energy-conserving Discontinuous Galerkin Meth-ods for the Vlasov-Ampere System
We propose energy-conserving numerical schemes for theVlasov-Ampere (VA) systems. The VA system is a model
PD13 Abstracts 97
used to describe the evolution of probability density func-tion of charged particles under self consistent electric fieldin plasmas. It conserves many physical quantities, includ-ing the total energy which is comprised of the kinetic andelectric energy. Unlike the total particle number conserva-tion, the total energy conservation is challenging to achieve.For simulations in longer time ranges, negligence of thisfact could cause unphysical results, such as plasma selfheating or cooling. In our work, we develop the first Eu-lerian solvers that can preserve fully discrete total energyconservation. The main components of our solvers includeexplicit or implicit energy-conserving temporal discretiza-tions, an energy-conserving operator splitting for the VAequation and discontinuous Galerkin finite element meth-ods for the spatial discretizations. We validate our schemesby rigorous derivations and benchmark numerical exam-ples such as Landau damping, two-stream instability andbump-on-tail instability.
Yingda ChengDepartment of MathematicsMichigan State [email protected]
Andrew J. ChristliebMichigan State UniverityDepartment of [email protected]
Xinghui ZhongMichigan State [email protected]
MS42
The Fractional Laplacian Operator as a SpecialCase of the Nonlocal Diffusion Operator
We analyze a nonlocal diffusion operator having as a spe-cial case the fractional Laplacian operator. We demon-strate that the solution of the nonlocal equation convergesto the one of the fractional Laplacian as the nonlocal inter-actions become infinite. Through several numerical exam-ples we illustrate the theoretical results and we show thatby solving the nonlocal problem it is possible to obtainaccurate approximations of the solutions of fractional dif-ferential equations circumventing the problem of treatinginfinite-volume constraints.
Marta D’EliaDepartment of Scientific ComputingFlorida State University, Tallahassee, [email protected]
Max GunzburgerFlorida State [email protected]
MS42
Homogenization of the Nonlocal Navier System ofEquations
Abstract not available at time of publication.
Tadele MengeshaPenn State UniversityDepartment of Mathematics
MS42
Finite Element Calculations in One-DimensionalPeridynamics and Multiscale Mono-models
The peridynamics theory of solid mechanics is a general-ization of classical continuum mechanics suitable for thesimulation of material failure and damage. As a contin-uum model, peridynamics can be discretized through manydifferent schemes. However, most of its current implemen-tations use the so-called meshfree discretization, which re-duces the governing equation to a discrete equation rep-resenting the dynamics of a system of virtual particlesinteracting in space; this discretization method is rela-tively inexpensive and suitable for simple implementationsof bond-breaking criteria yet introduces important limita-tions on the choice of horizon, affecting the potential useof the theory in multiscale modeling. It is thus of inter-est to explore other possible discretization schemes. Thispresentation will be focused on the finite element methodfor peridynamics, applied to one-dimensional models. Wewill present element by element calculations and comparenonlocal results with their local counterparts based uponthe finite element method for classical partial differentialequations. We present analytical calculations for a certainchoice of peridynamic kernel and demonstrate that the non-locality of the peridynamic model depends upon the ratioof the horizon and the finite element mesh size. This re-sult suggests a multiscale approach based upon differentdiscretizations of a single model, which we referred to as amultiscale mono-model. Analytical results are supportedby numerical experiments.
Pablo SelesonThe University of Texas at [email protected]
MS43
Homogenization of High-Contrast Brinkman Flows
Porous media flow can have many scales and is often amulti physical processes. A useful tool in the analysis ofsuch problems in that of homogenization as an averageddescription is derived circumventing the need for compli-cated simulation of the fine scale features. In this talk, werecall recent developments of homogenization techniques inthe application of flows high contrast media. We have par-ticular emphasis on the Brinkmann equation and relatednumerical methods.
Donald BrownKing Abdullah University of Science and TechnologyCenter for Numerical Porous [email protected]
Guanglian LiTexas A&M [email protected]
Yalchin EfendievDept of MathematicsTexas A&M [email protected]
Viktoria SavatorovaN6 NRNU MEPhI
98 PD13 Abstracts
MS43
Error Estimates for Mesoscale Continuum Modelsof Particle Systems
Abstract not available at time of publcation.
Alexander PanchenkoWashington State [email protected]
MS43
Modeling Effective Acoustic Behavior of PorousMedia with Microstructure
In this talk we present some recent results in mathemat-ical modeling of porous media with multi-scale structure.In particular, we focus on effective acoustic properties ofcancellous bone. Using two-scale convergence and other ho-mogenization tools we derive effective material propertiesof the fluid-filled porous matrix. The effective equationsare Biot-like, however they account for possible effects ofmicro-structural anisotropy.
Ana VasilicUAE [email protected]
Robert P. GilbertDepartment of Mathematical SciencesUniversity of [email protected]
Alexander PanchenkoWashington State [email protected]
MS43
Effect of Slip Boundary Conditions on the EffectiveBehavior of Suspensions and Membranes
The first to depart from no-slip conditions for viscous flu-ids on a solid surface was Navier (1827) who imposed a slipcondition that gives a linear dependence of the tangentialvelocity on the tangential stress. Threshold slip was intro-duced for viscous flow problems by Fujita (1994) who stud-ied the existence and uniqueness for the Stokes problemwith boundary conditions of the type −(σN)τ ∈ g ∂|uτ |,where g ≥ 0. A similar threshold-type condition was in-troduced by Fujita to model leaky boundaries: the normalvelocity is zero unless the jump in the normal stress acrossthe membrane reaches an yield limit. We study the macro-scopic effect of threshold slip for a suspension of rigid par-ticles in a viscous fluid, and that of threshold leak for apermeable membrane. For suspensions, the Navier-Fujitatype slip introduces at the macroscale a new fluid viscos-ity. For membranes, the effective boundary conditions areof subgradient type with an effective yield limit, in the caseof a densely distributed solid part, or of Navier type, in thecase of dilute solid part; in the intermediate case the tan-gential slip cancels, whereas the normal velocity and stressare continuous. Unlike in the case of perforated walls, nostress concentrations are present.
Bogdan M. VernescuWorcester Polytechnic InstDept of Mathematical Sciences
MS44
On Incompressible Flows with Singular Velocities
The purpose of the talk is to establish several existence,uniqueness and ill-posedness results for two families of ac-tive scalar equations with velocity fields determined by thescalars through very singular integrals. The first family isa generalized surface quasi-geostrophic equation with thevelocity field u related to the scalar θ by u = ∇⊥Λα−1θ,where 0 < α ≤ 1 and Λ = (−Δ)1/2 is the Zygmundoperator. The borderline case α = 0 corresponds tothe surface quasi-geostrophic equation and the situationis more singular for α > 0. We obtain the local exis-tence and uniqueness of classical solutions, the global ex-istence of weak solutions and the local existence of patchtype solutions. The second family is a singularly modifiedversion of the 2D incompressible porous media equationwith the velocity field v given by the scalar ρ as follows:v = −∇Λ−2+α∂x2ρ − (0,Λαρ), for 0 < α ≤ 1. Here, thecase α = 0 yields Darcy’s law. In contrast, for the last sin-gular active scalar equation local well-posedness does nothold for smooth solutions, but it does hold for certain weaksolutions.
Francisco GancedoUniversity of [email protected]
MS44
Big Box Limit for NLS
I will describe the derivation of a new equation from NLSon the torus with size L, in the weakly nonlinear regime,as L goes to infinity. I will then discuss various propertiesof this equation.
Pierre GermainNew York [email protected]
Zaher [email protected]
Erwan FaouINRIA [email protected]
MS44
Persistency of the Spatial Analyticity for NonlinearWave Equations and the Cubic Szego Equation
Gevrey classes were introduced by Maurice Gevrey in 1918to generalize real analytic functions. Functions of Gevreyclasses can be characterized by an exponential decay oftheir Fourier coefficients. This characterization has beenproved useful for studying analytic solutions of various non-linear PDEs, since the work by Foias and Temam (1989)on the Navier-Stokes equations. We use this technique toinvestigate the persistency of spatial analyticity for nonlin-ear wave equations (joint work with Edriss S. Titi), and thecubic Szego equation (joint work with Patrick Gerard andEdriss S. Titi). An advantage of this method is that it pro-vides a lower bound for the radius of the spatial analyticity
PD13 Abstracts 99
of the solutions.
Yanqiu GuoThe Weizmann Institute of [email protected]
MS44
Sensitivity Analysis with Respect to Principal Pa-rameters of Long Time Dynamics in HyperbolicSystems with Critical Exponents
We consider a third order in time equation which arises asa model of wave propagation in viscous thermally relax-ing fluids. The nonlinear PDE model under considerationdisplays two important characteristics: (i) it is quasilinearand (i) it is degenerate in the principal part. The main goalof this talk is to present results on existence-nonexistenceand decay rates for the solutions as a function of physi-cal parameters in the equation. This is joint work withBarbara Kaltenbacher, University of Graz.
Irena M. LasieckaUniversity of [email protected]
MS45
Finite Element Exterior Calculus Methods for Geo-physical Fluid Dynamics
We show how mixed finite element methods that fall intothe finite element exterior calculus framework can be ap-plied to the design of numerical weather prediction models.This investigation is being driven by the need to extendthe properties of the C-grid finite difference staggering toarbitrary triangular/quadrilateral meshings of the spherewhich are required for scalable massively parallel weatherforecasting. We shall discuss the following: (a) Poissonstructure, Casimirs, (b) geometric forms of stabilisation ofthe discretisation, (c) conservation laws, (d) treatment oncurved elements, (e) extension to fully three dimensionalmodels.
Colin CotterImperial College LondonDepartment of [email protected]
MS45
Numerical Approximation of Boussinesq-Green-Naghdi Models for Near-Shore Wave Physics
Boussinesq-Green-Naghdi equations model the physics ofwaves in the near shore. They are approximations to thefull Navier-Stokes equations where one assumes a verticalprofile for the wave velocity. The resulting model is a highlynonlinear system of PDES with higher order derivatives, al-beit in one less space dimension than the full model. Themodel is well-suited for discontinuous Galerkin approxi-mations. We describe a general approach and discuss theaccuracy and stability of the numerical approximation, aswell as validation of the model against experimental data.
Clint DawsonInstitute for Computational Engineering and SciencesUniversity of Texas at [email protected]
Nishant PandaThe University of Texas at Austin
MS45
Discontinuous Petrov Galerkin Methods for WavePropagation
Discontinuous Petrov Galerkin (DPG) methods use stan-dard trial spaces and nonstandard test spaces. The testspaces are automatically computed to obtain good stabil-ity constants. The resulting method can also be interpretedas a least-squares finite element method that minimize aresidual in a nonstandard norm. In this talk, we reportour results obtained by applying the DPG methodology totime-harmonic acoustic wave propagation. The main ad-vantage of the method is that it yields Hermitian positivedefinite systems which are easier to solve. Other least-squares methods also have the same advantage, but weshow how DPGmethods improve on standard least-squaresmethods.
Jay GopalakrishnanPortland state [email protected]
MS45
DGSEM-ALE Approximation of Reflection andTransmission at Moving Interfaces
We derive a spectrally accurate moving mesh methodfor mixed material interface problems for Maxwell’s andthe classical wave equation. We use a discontinuousGalerkin spectral element approximation with an arbitraryLagrangian-Eulerian mapping and derive the exact upwindnumerical fluxes that model the physics of wave reflectionand transmission at jumps in material properties. Spec-tral accuracy is obtained by placing moving material in-terfaces at element boundaries and solving the appropri-ate Riemann problem. We present examples showing theperformance of the method for plane wave reflection andtransmission at dielectric and acoustic interfaces.
Andrew Winters, David A. KoprivaDepartment of MathematicsThe Florida State [email protected], [email protected]
MS46
Stability Analysis of Flock Rings for 2nd OrderModels in Swarming
In this work we consider second order models in swarm-ing in which individuals interact pairwise with a power-lawrepulsive-attractive potentials. We study the stability forflock ring solutions and we show how the stability of thesesolutions is related to the stability of a first order model. Inunstable situations it is also possible to observe formationof clusters and fat rings.
Daniel BalagueDepartament de MatematiquesUniversitat Autnoma de [email protected]
Giacomo AlbiDipartimento di Matematica e InformaticaUniversita di [email protected]
100 PD13 Abstracts
Jose CarrilloDepartment of MathematicsImperial College [email protected]
James von BrechtUniversity of California, Los [email protected]
MS46
Two Short Stories on Existence and Stability of N-Vortex Equilibria in Fluids
Abstract: Vortex relative equilibria are configurations ofvortices that maintain their basic shapes for all time, whilerotating or translating rigidly in space. It is common toobserve these patterns in nature and experiments, wherethe underlying dynamics are governed, for example, by thetwo-dimensional Euler or Navier-Stokes equations. Onecan often take advantage of the structure of vortices toderive equations of motion that are much simpler than theoriginal PDE. In this work, we look at two point vortexmodels to study the existence and stability of a numberof relative equilibria. We complement this analysis withcomparisons of our results to crystals observed in variousexperiments and numerical simulations.
Anna M. BarryInstitute for Mathematics and its ApplicationsUniversity of [email protected]
MS46
Bifurcation Dynamics in a Nonlocal HyperbolicModel for Self-organised Biological Aggregations
The investigation of complex spatial and spatio-temporalpatterns exhibited by self-organised biological aggregationshas become a very active research area. Here, we focus ona class of nonlocal hyperbolic models for self-organised ag-gregations, and investigate the bifurcation dynamics of avariety of spatial and spatio-temporal patterns observednumerically near codimension-1 and codimension-2 bifur-cation points.
Raluca EftimieUniversity of Alberta, Edmonton, [email protected]
MS46
Vortex Crystals, Animal Skin Patterns and IceFishing
I will discuss three very different topics which turn out tohave a related mathematical formulation. The first topic isclassical vortex dynamics in the plane. We look for relativeequilibria (lattice crystals) in the limit of large number ofvortices. Many results for the steady state and its localstability can be obtained taking the mean-field limit. Next,we consider hot-spot solutions to certain reaction-diffusionPDE system. Similar PDE’s are often used to model spotson the animal skins. We derive a reduced system of ODE’swhich describe the motion and locations of these spots.In certain regimes, we show that these ODE’s are relatedto the relative equlibria of the vortex model. Finally, wedescribe the problem of mean first passage time. Supposeyou live in Canada and you want to catch a fish in a lakecovered by ice. Where do you drill a hole to maximize your
chances? This question can be reformulated in terms of arandom walk of brownian particle; the answer is given interms of a certain optimization problem. Its solution isagain related to vortex crystals. Joint work with: YuxinChen, Daniel Zhirov and Michael Ward
Theodore KolokolnikovDalhousie [email protected]
MS47
Analysis of Point Defects in a Ferroelectric Liq-uid Crystal Using a Generalized Ginzburg-LandauModel
Defects in Smectic C* liquid crystal film cause distinc-tive spiral patterns in the texture of the film. This par-ticular ferroelectric thin film is modeled by a generalizedGinzburg-Landau energy functional,
1
2
∫Ω
k1(div u)2 + k2(curl u)2 +
1
2ε2(1− |u|2)2 dx.
Given fixed boudary data of degree d > 0, we study thelimiting configuration of a sequence of minimizers {uε} asε → 0. We show the limit function contains d degree onesingularities and near each, the function asymptotically haseither a bend or splay pattern, depending on the relativevalues of k1 and k2. We also show that this functional has arenormalized energy that is minimized by the singularitiesof the limit function.
Sean A. Colbert-KellyNational Institute of Standards and [email protected]
Daniel PhillipsDepartment of Mathematics, Purdue [email protected]
Geoffrey McFaddenNational Institute of Standards and [email protected]
MS47
Electric-Field-Induced Instabilities in Liquid-Crystal Films
Simple models of liquid crystals characterize orientationalproperties in terms of a director field, which can be influ-enced by an applied electric field. The local electric field isinfluenced by the director field; so equilibria must be com-puted in a coupled way. Equilibria are stationary points ofa free energy functional, which can fail to be coercive be-cause of the nature of the director/electric-field coupling.We discuss characterizations of local stability and anoma-lous behavior in such systems.
Eugene C. GartlandKent State [email protected]
MS47
Regularity Properties for Nematic Liquid Crystals
We investigate regularity properties for and bounds on lo-cal minimizers to an energy for nematic liquid crystals with
PD13 Abstracts 101
a Maier-Saupe potential.
Patricia BaumanPurdue [email protected]
Daniel PhillipsDepartment of Mathematics, Purdue [email protected]
MS48
Free-boundary Problems in Surface Evolution byCurvature Flows
We will discuss the qualitative properties of solutions tofully non-linear parabolic equations that arise from the evo-lution of hyper-surfaces by functions of their principal cur-vatures. Such examples include evolutions by powers of theGaussian curvature or evolution by the harmonic mean ofthe principal curvatures. Since such equations become de-generate or singular the classical regularity results fail. Wewill discuss the optimal regularity of solutions and relatedfree-boundary problems.
Panagiota DaskalopoulosColumbia [email protected]
MS48
A Classical Perron Method for Existence of SmoothSolutions to Boundary Value and Obstacle Prob-lems for Degenerate-elliptic Operators via Holo-morphic Maps
We prove existence of solutions to boundary value prob-lems and obstacle problems for degenerate-elliptic, lin-ear, second-order partial differential operators with partialDirichlet boundary conditions using a new version of thePerron method. The elliptic operators considered have adegeneracy along a portion of the domain boundary whichis similar to the degeneracy of a model linear operatoridentified by Daskalopoulos and Hamilton (1998) in theirstudy of the porous medium equation or the degeneracyof the Heston operator (Heston, 1993) in mathematical fi-nance. Our Perron method relies on weak and strong max-imum principles for degenerate-elliptic operators, conceptsof continuous subsolutions and supersolutions for bound-ary value and obstacle problems for degenerate-elliptic op-erators, and maximum and comparison principle estimatespreviously developed by the author (Feehan, 2012).
Paul FeehanRutgers [email protected]
MS48
Optimal Regularity and the Free Boundary in theParabolic Signorini Problem
We give a comprehensive treatment of the parabolic Sig-norini problem based on a generalization of Almgren’smonotonicity of the frequency. This includes the proofof the optimal regularity of solutions, classification of freeboundary points, the regularity of the regular set and thestructure of the singular set.
Donatella DanielliPurdue [email protected]
Nicola GarofaloUniversita’ di [email protected]
Arshak PetrosyanPurdue UniversityDepartment of [email protected]
Tung ToPurdue [email protected]
MS48
Regularity of Solutions and of the Free Boundary ofthe Obstacle Problem for the Fractional Laplacianwith Drift
We study the elliptic obstacle problem defined by the frac-tional Laplacian with drift,
min{Lu(x), u(x)− ϕ(x)} = 0, ∀x ∈ Rn,
where the operator L is defined by
Lu(x) = (−Δ)su(x) + b(x) · ∇u(x) + c(x)u(x), ∀x ∈ Rn.
In joint work with Arshak Petrosyan, we develop a newmonotonicity formula and we establish the optimal regu-larity of solution, C1,s(Rn), and Lipschitz regularity of thefree boundary in neighborhoods of regular points, in thecase 1/2 < s < 1. In the case 0 < s < 1/2, the symbol as-sociated to the operator L, a(x, ξ) = |ξ|2s + ib(x) · ξ+ c(x),is no longer elliptic. In joint work with Charles Epstein, weprove that if Lu ∈ Hk(Rn), for some real constant k, and
u is a tempered distribution, then u belongs to Hk+2sloc (Rn)
and b(x) · ∇u(x) belongs to Hkloc(R
n).
Camelia A. PopRutgers [email protected]
Arshak PetrosyanPurdue UniversityDepartment of [email protected]
Charles EpsteinUniversity of [email protected]
MS49
Fluid-Structure Interaction Problems in OcularBlood Flow
Ocular hemodynamics plays a crucial role in the patho-physiology of several ocular diseases, including glaucoma,age-related macular degeneration and diabetic retinopathy.The mathematical modeling of ocular blood flow involvesthe motion of a fluid (blood) through deformable elasticstructures. In this talk, we will present a model describ-ing the blood flow through the lamina cribrosa, a collagenstructure structure that is pierced by the central retinalvessels approximately in its center and maintains the pres-sure difference between the intraocular pressure inside theeye globe and the retrolaminar tissue pressure inside theoptic nerve canal. The analytical properties of the model,
102 PD13 Abstracts
based on the theory for poroelastic materials, will be dis-cussed and its numerical solutions will be presented forvarious clinically-relevant scenarios.
Giovanna GuidoboniIndiana University-Purdue University at IndianapolisDepartment of Mathematical [email protected]
MS49
Intrinsic Decay Rates for the Energy of Second Or-der Nonlinear Evolutions with Viscoelasticity
Nonlinear second order evolutions subject to viscoelasticdamping are considered. It is well known that the stabilitycharacteristics of finite energy solutions are dependent onthe decay properties of the relaxation kernel. The aim ofthe talk is to present sharp decay rates of solutions corre-sponding to relaxation kernels without quantified asymp-totic behavior. The obtained decay rates are described bysolutions to appropriately constructed ordinary differentialequations. Applications to nonlinear waves and von Kar-man systems of dynamic elasticity with memory will begiven.
Irena M. LasieckaUniversity of [email protected]
MS49
Model Development and Uncertainty Quantifica-tion for Systems with Nonlinear and Hysteretic Ac-tuators
Smart material actuators and sensors provide unique con-trol capabilities for a range of applications involving fluid-structure and flow-structure interactions. These includePZT-based macrofiber composites which are being consid-ered for flow control and shape memory alloys which are be-ing tested for use as catheters employed for laser treatmentsof atrial fibrillation. In this presentation, we will discussthe development of a modeling framework for these systemsthat facilitates subsequent design, uncertainty quantifica-tion, and real-time control implementation for transducersoperating in highly nonlinear and hysteretic regimes.
Ralph C. SmithNorth Carolina State UnivDept of Mathematics, [email protected]
MS49
New Advances and Open Problems in NonlinearFlow-Plate Interactions
We address nonlinear PDE models arising in the studyof flow-plate interactions and the associated phenomenon(e.g. flutter). We will review the recent well-posedness andlong-time behavior results for a clamped, non-rotationalplate immersed in a supersonic flow. Additionally, we ad-dress novel mathematical challenges associated to the well-posedness theory for other physical configurations of re-cent experimental and numerical interest: specifically, theKutta Joukowsky flow condition and free plate boundaryconditions.
Justin Webster, Irena M. LasieckaUniversity of Virginia
[email protected], [email protected]
MS50
Regularity of Solutions to the Axisymmetric EulerEquations
We investigate properties of solutions to the axisymmetricEuler equations when initial vorticity belongs to a space ofBesov type. In particular, we study regularity of solutionswhen a certain Besov norm of initial vorticity diverges ina controlled way.
Elaine CozziDept. Math - Oregon State [email protected]
MS50
Holder Continuous Euler Flows
Motivated by the theory of hydrodynamic turbulence, LarsOnsager conjectured in 1949 that solutions to the incom-pressible Euler equations with Holder regularity below 1/3may fail to conserve energy. I will discuss Onsager’s conjec-ture and recent progress towards this conjecture, includingthe construction of solutions to Euler which are (1/5− ε)-Holder continuous and fail to conserve energy.
Phil IsettDept. Math - Princeton [email protected]
MS50
Nonlinear Stability of Vortex Pairs
In this talk, we discuss nonlinear orbital stability forsteadily translating vortex pairs, a family of nonlinearwaves that are exact solutions of the incompressible, two-dimensional Euler equations. We use an adaptation ofKelvin’s variational principle, maximizing kinetic energypenalised by a multiple of momentum among mirror-symmetric isovortical rearrangements. This formulationhas the advantage that the functional to be maximizedand the constraint set are both invariant under the flow ofthe time-dependent Euler equations, and this observationis used strongly in the analysis. Previous work on existenceyields a wide class of examples to which our result applies.
Geoffrey BurtonUniv. of [email protected]
Milton Lopes Filho, Helena J. Nussenzveig LopesUniversidade Federal do Rio de [email protected], [email protected]
MS51
Algebraic Vortex Spirals
Vortex spirals are ubiquitous in fluid flow, for example asturbulent eddies or as trailing vortices at aircraft wings.However, there are few proofs of existence for any of thecommon fluid models. We consider solutions of the in-compressible Euler equations that have vorticity stratify-ing into algebraic spirals. The solutions are selfsimilar:velocity v(t, x) = tm−1v(t−mx), for similarity exponent12
< m < ∞. Selfsimilar flows are special solutions ofthe full initial-value problem, but obtained by solving more
PD13 Abstracts 103
tractable boundary value problems. The key to the exis-tence proof is an coordinate change which is implicit, de-pending on the a priori unknown solution. We will alsodiscuss the importance of the program for showing non-uniqueness in the initial-value problem for the 2d incom-pressible Euler solutions.
Volker W. EllingUniversity of [email protected]
MS51
Linear and Nonlinear Reflection Patterns in GasDynamics
The analysis of self-similar, non-potential flow for the equa-tions of gas dynamics involves the interaction of the non-linear family of acoustic waves with the linear, degeneratefamily or families of shear (and possibly entropy) waves.We examine a simple pattern, regular reflection of a shockat wedge near the reflection point. Behind a transonicshock, the self-similar equations change type, with theacoustic waves now corresponding to a second-order ellipticequation coupled with a hyperbolic component correspond-ing to the shear waves (in isentropic flow). The ellipticsolution operator is compact, while the transport systemgives rise to a contraction. We show how to couple theseto produce a local solution.
Barbara KeyfitzThe Ohio State [email protected]
Katarina JegdicUniversity of [email protected]
Suncica CanicDepartment of MathematicsUniversity of [email protected]
Hao YingThe Ohio State [email protected]
MS51
Hyperbolic Techniques for Multidimensional Inter-face Dynamics
We are interested in interfacial dynamics in continuum me-chanics which are governed by hyperbolic evolution equa-tions in the bulk and appropriate transfer conditions acrossthe interface. Instances of bulk systems are providedby compressible Euler equations for liquid-vapour flow,nonlinear elasticity for solid-solid transitions, or Buckley-Leverett type formulations for in porous media flow. Trans-fer conditions have not only to gurantee conservation prop-erties but account for e.g. curvature, phase transition orconnect different types of bulk physics. To track the inter-faces’ local dynamics IVPs in normal directions are consid-ered. This leads to generalized Riemann problems whichare hardly to solve exactly. We will introduce approxi-mations which rely on regularization or hyperbolic-ellipticrelaxation. The convergence of the approximations and the(non)validity of entropy conditions are discussed.
Christian RohdeUniversity of Stuttgart
Institut fur Angewandte Analysis und [email protected]
MS51
The Onset of Cavitation for the Equations of Poly-convex Elasticity
Cavitation refers to the opening of holes during the dy-namic deformation of an elastic material and is a phe-nomenon that lies at the boundary of continuum model-ing. We will describe a solution constructed by Pericak-Spector and Spector describing a cavity opening from ahomogeneously deformed state. This poses challenges tothe existence theory of multi-d conservation laws, as theconstructed cavitating solution turns out to decrease themechanical energy. We will analyze this example from theperspective of a theory that accounts for the singular layersof the cavitating solution. We call the resulting notion ofsolution as singular limiting induced from continuum so-lution. It turns out that this perspective can account forthe cost of energy associated with producing the cavity,and that the resulting energy of the cavitating solution islarger than that of a homogeneously deformed state.
Athanasios TzavarasUniversity of [email protected]
MS52
Asymptotic Approximation of the Dirichlet to Neu-mann Map of High Contrast Conductive Media
I will present an asymptotic study of the Dirichlet to Neu-mann map of high contrast composite media with perfectlyconducting inclusions that are close to touching. The resultis an explicit characterization of the map in the asymptoticlimit of the distance between the particles tending to zero.
Liliana BorceaRice UniversityDept Computational & Appl [email protected]
MS52
Anomalous Diffusion of a Tracer Particle in FastCellular Flow
It is well known that a diffusive tracer particle in the pres-ence of an array of strong opposing vortices (aka cellularflow) behaves like an effective Brownian motion on longtime scales. On intermediate time scales, however, a ro-bust anomalous diffusive behaviour has been numericallyobserved. This talk is a first step towards understandingthis anomalous behaviour. We will show that the varianceof the particle behaves like O(
√t) on ”intermediate” time
scales; in contrast, the long time behaviour of the varianceis like O(t).
Gautam IyerCarnegie Mellon [email protected]
MS52
Asymptotic Expansion for Wave Propagation in aMedia with a Small Hole
In this talk, I will discuss the asymptotic expansion of the
104 PD13 Abstracts
acoustic field under the effect of a small hole in the quasi-static regime. As a consequence, one can prove that thegradient of the field is ”bounded”. This is joint work withReitich and Lin.
Hoai Minh NguyenUniversity of MinnesotaSchool of [email protected]
MS52
Title Not Available at Time of Publication
Abstract not available at time of publication.
Lenya RyzhikStanford [email protected]
MS53
Tight-Binding Approximations and Edge States inHoneycomb Optical Lattices
Over the last several years, a great deal of interest hasemerged over honeycomb optical lattices, which are mod-eled by a Gross-Pitaevskii (GP) equation with a periodic,two-dimensional potential. Using a tight-binding approxi-mation in the semi-classical limit, a two-dimensional non-linear discrete system has been derived as an approxima-tion to the GP equation. We present results that establishthe validity of this approximation on asymptotically longtime scales. By introducing an edge into the discrete sys-tem, we then present results on the propagation of modeslocalized along the edge, or edge modes. Ignoring nonlin-earity, one can find a plethora of linear edge modes, buta central question is what is the impact of nonlinearity onthese problems. In the case of weak nonlinearity, we havedeveloped a rigorously justified approximation describingthe impact of nonlinearity on the linear modes. The mostimportant result from this is that the nonlinearity doesnot cause delocalization away from the edge. In the caseof strong nonlinearity, we present numerical results whichshow the nonlinearity does not cause delocalization, andthus edge modes should be a stable feature for optical lat-tices with edges.
Chris CurtisDepartment of Applied MathematicsUniversity of Colorado at [email protected]
MS53
Elastic Cloaking for Civil Structures
The talk will present some mathematical framework forelastic cloaking and will be followed by some numericalillustration and an experiment on seismic metamaterial.Such concepts have been developed in conjunction with theearlier works by Milton, Briane Willis (cloaking preserv-ing the symmetry of the elasticity tensor), Norris (acousticcloaking with pentamode materials) and Brun, Guenneau,Movchan (cloaking with an elasticity tensor without theminor symmetries). The large-scale seismic test held on asoil metamaterial using vibrocompaction probes is the firstapplication of cloaking to civil engineering. Joint work withS. Enoch, A. Diatta, S. Brule and E. Javelaud.
Sebastien GuenneauInstitut Fresnel, UMR CNRS 6133
MS53
Engineering Anisotropy to Amplify a Long Wave-length Field Without a Limit
We present a simple design of circular or spherical shells ca-pable of amplifying a long wavelength or static field. Thisdesign makes use of only two isotropic materials and isoptimal restricted to the prescribed geometric and materi-als constraint. Further, it is shown that the amplificationfactor of the structure can be made arbitrarily large asthe ratio of the radius of the inner sphere to that of theouter sphere decreases to zero. It is anticipated that thepresented design will be useful for high gain antenna fortelecommunication, magnets generating strong, local uni-form fields, and thermoelectric devices harvesting thermalenergy.
Liping LiuDepartment of Math, Mechanical EngineeringRutgers [email protected]
MS53
Pathological Scattering in the Resonant Slow-LightRegime
Layered media consisting of alternating magnetic andanisotropic components were shown by Figotin and Viteb-skiy to admit an electromagnetic mode of zero energy fluxat a frequency in the interior of a propagation band; oper-ation near this frequency is called the “slow-light” regime.We consider scattering of waves by a planar defect em-bedded in a slow-light ambient medium. The scatteringproblem is pathological because three eigenmodes coalesceinto the one zero-flux mode as branches of a Puiseux seriesin a perturbation parameter ?, and the completeness andindependence of the rightward and leftward modes breaksdown. It is even more pathological when the defective layeradmits a guided resonant mode exactly at the slow-lightparameters (?=0). The unusual scattering phenomena fitinto a perturbative linear-algebraic setting; the players area fixed indefinite flux form in C4, an analytic flux-unitarytransfer matrix T(?), and an analytic flux-selfadjoint ma-trix K(?), where K(0) has a non-trivial Jordan block of size3. Joint Work with Aaron Welters, MIT.
Stephen P. ShipmanDepartment of MathematicsLouisiana State [email protected]
MS54
On Forced Turbulence in Physical Scales of 3D NSE
We apply the physical scales framework to prove that Kol-mogorov Dissipation Law ε ∼ U3/L implies energy cascadein the 3D periodic NSE, provided the force is large enough.We also obtain restrictive scaling properties of the solutionsin this case, and compare our results to the ones obtainedin the classical Fourier framework.
Radu DascaliucOregon State [email protected]
Zoran Grujic
PD13 Abstracts 105
University of [email protected]
MS54
The Well-posedness Linear Hyperbolic Partial Dif-ferential Equations in a Rectangle
In this article, we consider the linear hyperbolic InitialBoundary Value Problems (IBVP) in a rectangle in boththe constant and variable coe cients cases. We use semi-group method instead of Fourier analysis to achieve thewell-posedness of the linear hyperbolic system, and we findby diagonalization that there are only two simple modes,which we call hyperbolic and elliptic modes in the system.The hyperbolic system in consideration is either symmetricor Friedrichs-symmetrizable. We also give some applica-tions of our results.
Aimin HuangIndiana University BloomingtonDepartment of [email protected]
Roger M. TemamInst. for Scientific Comput. and Appl. Math.Indiana [email protected]
MS54
An Appropriate Notion of Attractor for Semi-Dissipative Equations
The large-time behavior of a dissipative system can oftenbe understood by studying its global attractor, which cancontain deep information about its underlying structure.We will show that the notion of attractor can be extendedto systems with only partial dissipation, and apply the con-cept to fluid dynamics. We will see that this generalizedattractor not only has a rich structure, but also encodes awealth of turbulent phenomena in a single object.
Animikh BiswasUniversity of North Carolinaat [email protected]
Ciprian FoiasTexas A & M,Indiana [email protected]
Adam LariosDepartment of MathematicsTexas A&M [email protected]
MS54
Existence and Uniqueness of Solutions for the In-viscid Shallow Water Equations
Motivated by the equations of the large scale oceans and at-mosphere (primitive equations), we discuss the issue of ex-istence and uniqueness of solutions for the linearized shal-low water equations in space dimension two in a rectangle.We also study the nonlinear shallow water equations insome subcritical and supercritical situations. The choiceof the suitable boundary conditions and the fact that thedomain (rectangle) is not smooth, are two essential issues
in this study. In particular we show how suitable boundaryconditions make the initial and boundary problem mildlydissipative and well-posed.
Roger M. TemamInst. for Scientific Comput. and Appl. Math.Indiana [email protected]
Aimin HuangIndiana [email protected]
MS55
Nonstandard Dispersive Estimates and Long TimeExistence for the 2d Water Wave Problem
In this talk, I will presents recent results on the relationshipbetween the decay of a solution to the linearized waterwave problem and its initial data and implications for thelong time existence of gravity waves. Certain new decaybounds for a class of 1D dispersive equations (including thelinearized water wave) display a surprising growth factor,which we show is sharp. A further exploration leads to aresult relating singularities of the initial data at the originin Fourier frequency to the regularity of solutions to thesedispersive equations.
Jennifer BeichmanUniversity of [email protected]
Sijue WuUniversity of Michigan at Ann [email protected]
MS55
Finite Time Blow-Up Versus Global Existence ofSolutions for a 1D Fluid Model
We discuss the finite time blow-up versus global well-posedness for a 1D fluid model. This model may be de-rived by reducing the Navier-Stokes equations, the prim-itive equations, Burgers equations and many other fluidmodels. This is a joint work with K. Nakanishi and E. S.Titi.
Slim IbrahimUniversity of [email protected]
MS55
Scattering for Small Solutions of NLS with Sub-critical Nonlinearity
We prove global existence and scattering for small scaleinvariant data for a model close to the standard cubic NLS.The proof is based on the method of normal forms. Thisis joint work with Vladimir Georgiev, University of Pisa.
Atanas StefanovUniversity of [email protected]
MS55
Justifying the Modulation Approximation of the
106 PD13 Abstracts
Full Water Wave Problem
In this joint work with Sijue Wu (U. Mich.), we considersolutions to the inviscid infinite depth water wave problemneglecting surface tension which are to leading order wavepackets with small amplitude and slow spatial decay thatare balanced. It has been known formally since Zakharovin the 60s that such solutions have modulations that evolveaccording to a focusing cubic NLS equation on very slowtime scales. Justifying this rigorously is a real problem,since standard existence results for the water wave prob-lem do not yield solutions which exist for long enough tosee the NLS dynamics. Nonetheless, given initial data closeto a wave packet in a suitable L2 Sobolev space, we showthat there exists a unique solution to the water wave prob-lem which is close to the formal approximation for appro-priately long times. This is done by applying the energymethod to formulations of the evolution equations for thewater wave problem in 2D and 3D developed by Sijue Wuwith quadratic nonlinearities that are either absent or insome sense mild.
Nathan TotzDepartment of Mathematics, Duke [email protected]
MS56
On a Chemotaxis Model with Saturated Chemo-tactic Flux
We propose a PDE chemotaxis model, which can be viewedas a regularization of the Patlak-Keller-Segel (PKS) sys-tem. Our modification is based on a fundamental physicalproperty of the chemotactic flux function – its bounded-ness. This means that the cell velocity is proportional tothe magnitude of the chemoattractant gradient only whenthe latter is small, while when the chemoattractant gradi-ent tends to infinity the cell velocity saturates. Unlike theoriginal PKS system, the solutions of the modified modeldo not blow up. After obtaining local and global existenceresults, we use the local and global bifurcation theories toshow the existence of one-dimensional spiky steady states;we also study the stability of bifurcating steady states. Fi-nally, we numerically verify these analytical results. Jointwork with A. Kurganov, X. Wang and Y. Wu
Alina ChertockNorth Carolina State UniversityDepartment of [email protected]
MS56
G-Convergence of Elliptic Operators with Nonstan-dard Growth
We consider a class of monotone elliptic operators withnonstandard growth condition. We present a preliminaryresult on G-compactness under certain additional technicalassumptions.
Alexander PankovDepartment of MathematicsMorgan State University, [email protected]
MS56
Parabolic Equations Degenerating on a Part of the
Domain
In this work we study nonlinear parabolic equations of thep-Laplace type degenerating on a part of the domain. Thedegeneration is controlled by a small parameter. For thefixed value of this parameter, the results we discuss fallwithin the scope of E. DiBenedetto’s results. We obtainestimates uniform with respect to ”epsilon” , thus extend-ing the results, obtained earlier by Yu.A. Alkhutov andV.A. Liskevich in the linear case.
Mikhail D. SurnachevInsitute of applied mathematics,Russion Academy of Science,[email protected]
Mikhail D. SurnachevKeldysh Institute of Applied MathematicsRussion Academy of [email protected]
MS56
Some Recent Progress on the Study of Behavior ofSolutions of Degenerate Viscous Hamilton-JacobiEquations
I will give a summary on a new approach to study largetime behavior of solutions of degenerate viscous Hamilton-Jacobi equations. Then I will describe its applications tothe usual cases, the obstacle problems, and the fully non-linear cases.
Hung TranUniversity of [email protected]
MS57
Stationary States and Asymptotic Behaviour ofAggregation Models with Nonlinear Local Repul-sion
We consider a continuum aggregation model with nonlin-ear local repulsion given by a degenerate power-law diffu-sion with general exponent. The steady states and theirproperties in one dimension are studied both analyticallyand numerically, suggesting that the quadratic diffusion isa critical case. The focus is on finite-size, monotone andcompactly supported equilibria. We also investigate nu-merically the long time asymptotics of the model by simu-lations of the evolution equation. Issues such as metasta-bility and local/ global stability are studied in connectionto the gradient flow formulation of the model.
Martin BurgerUniversity of MuensterMuenster, [email protected]
Razvan FetecauSimon Fraser [email protected]
Yanghong HuangImperial College [email protected]
MS57
Modeling Selective Local Interactions with Mem-
PD13 Abstracts 107
ory
In this talk we present our recent results on stochastic par-ticle methods for describing the local interactions betweencyanobacteria. In these models, particles selectively chooseone of their neighbors as the preferred direction of motion.Memory is incorporated into the model by allowing persis-tence. One- and two-dimensional models are studied.
Doron LevyUniversity of [email protected]
MS57
Minimization of an Energy Defined via anAttractive-Repulsive Interaction Potential Relatedto Nonlocal Aggregation Models
In this talk I will consider the constrained minimiza-tion of an energy defined via an attractive-repulsive in-teraction potential. This energy is related to aggrega-tion models given by the equation ρt + ∇ · (ρ(−∇K ∗ ρ))where ρ is the density of the aggregation and K is theinteraction potential. Looking at the energy E(ρ) =∫ ∫
K(x− y)ρ(x)ρ(y)dxdy for interaction potentials of theform K(|x|) = |x|q/q − |x|p/p in the parameter regime−N < p < q, we will establish the existence of minimiz-ers under certain assumptions depending on the regime ofpowers p and q. Moreover, we will consider the binary den-sity version of this energy where ρ ∈ {0, 1} and commenton the global minimizers when q = 2 and p = 2−N . Thisis a joint work with R. Choksi and R. Fetecau.
Ihsan A. TopalogluMcGill UniversityCRM Applied Mathematics [email protected]
MS57
Title Not Available at Time of Publication
Abstract not available at time of publication.
James von BrechtUniversity of California, Los [email protected]
MS58
Mathematical Models for a Soap Opera
In this talk, I will put forth a new energy for the descrip-tion of large distortions of lipid bilayers, which constitutethe base component of cell membranes. This energy ac-counts for the soap-like liquid crystalline nature of lipid bi-layers and the coupling between the deformations of lipidmolecules and layers. The analogies between soap-like liq-uid crystals, with an infinite number of layers, and lipidbilayers, with only two layers, will be further discussed.
Raffaella De VitaEngineering Science and MechanicsVirginia [email protected]
Iain W. StewartUniversity of Strathclyde
MS58
Higher-Order Gradient Theories for Nematic Liq-uid Crystals
We explore the connection between mathematical theoriesdescribing defects in nematic liquid crystals and crystallinesolids. The parallels between these theories allow us topropose an alternative framework for modeling nematic de-fects.
Amit AcharyaCivil and Environmental EngineeringCarnegie Mellon [email protected]
Dmitry GolovatyThe University of AkronDepartment of [email protected]
MS58
Stability of Radially Symmetric Solution inLandau-De Gennes Theory of Liquid Crystals
Abstract not available at time of publication.
Valeriy SlastikovUniversity of BristolBristol, [email protected]
MS58
Chevron Structures in Liquid Crystal Films
Abstract not available at time of publication.
Lei Zhang ChengSchool of SciencesIndiana University [email protected]
MS59
An Interface Capturing Method for Simulating Dy-namic Gel-Fluid Interaction
Many problems in biology involve gels which are mixturescomposed of a polymer network permeated by a fluid sol-vent (water). In some applications, free boundaries sep-arate regions of gel and regions of pure solvent, resultingin a degenerate system. To overcome this difficulty, wedevelop a simple interface-capturing method that consistsof a regularization procedure to solve the two-fluid modelequations describing gels composed of two viscous fluids,in the situation in which the volume fraction of one fluidvanishes in part of the domain.
Jian DuDepartment of MathematicsFlorida Institute of [email protected]
Robert D. GuyMathematics DepartmentUniversity of California [email protected]
108 PD13 Abstracts
Aaron L. FogelsonUniversity of [email protected]
Grady B. WrightDepartment of MathematicsBoise State University, Boise [email protected]
James P. KeenerUniversity of [email protected]
MS59
Toward a Theory of Mutual Attractions and Re-pulsions of Floating Objects
The horizontal force experienced by two infinite parallelplates of possibly differing solid materials, situated verti-cally and partially immersed in an infinite fluid bath undergravity, is characterized in the context of classical surfacetension theory. Varying kinds of behavior are encountered,depending on contact angles and on plate separation. Theparticular case, in which the two contact angles on theplate sides directly facing each other are supplementary,leads to a striking limiting instability.
Robert FinnStanford [email protected]
MS59
Coupling Between Fluid Equations and Fabric Sur-face Model
Abstract not available at time of publication.
Xiaolin LiDepartment of Applied Math and StatSUNY at Stony [email protected]
MS59
Compressible Navier-Stokes System with Temper-ature Dependent Viscosity and Heat Conductivity
The full system of compressible Navier -Stokes equationsis the model for the compressible fluids with heat transfer.Particularly, Chapman-Enskog theory suggests the viscos-ity and heat conductivity of a fluid are powers of the tem-perature. Feireisl’s type weak solution is constructed forthis model.
weizhe zhangGeorgia Institute of [email protected]
Ronghua PanGeorgia [email protected]
MS60
Weak Solutions for an Incompressible, GeneralizedNewtonian Fluid Interacting with a Linearly Elas-tic Koiter Shell
In this talk I will present an existence result for the interac-
tion of an incompressible, generalized Newtonian fluid witha linearly elastic Koiter shell whose motion is restrictedto transverse displacements. The moving middle surfaceof the shell constitutes the mathematical boundary of thethree-dimensional fluid domain. I show that weak solu-tions exist as long as one can exclude selfintersections ofthe shell.
Daniel LengelerUniversity of [email protected]
MS60
A Constructive Existence Proof for a MovingBoundary Fluid-Multi-Layered Structure Interac-tion Problem
We consider a nonlinear, unsteady, moving-boundary prob-lem of parabolic-hyperbolic-hyperbolic type modeling theinteraction between a Newtonian fluid and an elastic struc-ture composed of two layers: a thin layer modeled by the1D wave equation, and a thick layer modeled by the 2Dequations of linear elasticity. The thin layer is in contactwith the fluid, thereby serving as a fluid-structure inter-face with mass. We will present a constructive proof of anexistence of a weak solution.
Boris MuhaDepartment of Mathematics, Faculty of ScienceUniversity of [email protected]
Suncica CanicDepartment of MathematicsUniversity of [email protected]
MS60
The Motion of the Rigid Body with Collisions in aBounded Domain. Global Solvability Result
We consider the problem of motion of a rigid body in an in-compressible viscous fluid, filling a bounded domain. Thisproblem was studied by many autors. They consideredclassical non-slip boundary conditions, which gave themvery PARADOXICAL result of no collisions of the bodywith the boundary of the domain. In this work we studywhen Navier slip conditions are prescribed on the boundaryof the body (instead of non-slip conditions).We prove forthis model the GLOBAL existence of weak solution, whichpermits COLLISIONS with the boundary of the domain.Itis joint work with N. Chemetov.
Sarka NecasovaMathematical Institute of Academy of SciencesCzech [email protected]
MS60
Global Existence and Nonlinear Stability for Fluid-structure Problems
I will discuss both viscous an inviscid fluid-structure inter-action problems which couple either the Navier-Stokes orEuler equations to fully nonlinear thin elastic shell struc-tures.
Steve ShkollerUniviversity of California, Davis
PD13 Abstracts 109
Department of [email protected]
MS61
Global Existence of Solutions to Fluid StructureModel with Moving Interface and Boundary Dissi-pation
Fluid structure interaction model comprising of a N-Sequation coupled to a 3-D system of dynamic elasticityis considered. The coupling occurs on an interface be-tween the two media and the model accounts for a freeboundary of the solid which is moving within the fluid.The moving interface is subjected to a frictional damp-ing. It is proved that solutions corresponding to sufficientlysmall initial data are unique and they exist globally. Thisresult is established with minimal regularity assumptionsimposed on the data and without any interior dissipationimposed on the structure. The only source of added dis-sipation is the interface (boundary) damping. This is, toour best knowledge, the first global existence result for fluidstructure interaction in the 3 dimensional space. This isjoint work with Michaela Ignatova (Stanford University),Igor Kukavica (University of Southern California) and Am-jad Tuffaha (The Petrolium Institute, Abu-Dhabi, UAE)
Irena M. LasieckaUniversity of [email protected]
MS61
Steady Compressible Navier-Stokes-Fourier Sys-tem with Temperature Dependent Viscosities
We consider the system of partial differential equations de-scribing the steady flow of a compressible heat conductingNewtonian fluid in a bounded three-dimensional domain.We assume the pressure law of the form p(�, ϑ) ∼ �γ + �ϑwith γ > 1 and the viscosities ∼ (1 + ϑ)α, α ∈ [0, 1]. Weshow the existence of a weak or variational entropy solu-tion for the above model in dependence on the parametersγ, α and the form of the heat flux.
Milan PokornyCharles University in [email protected]
MS61
Generalized Stokes System and Implicit Constitu-tive Relations
We are interested in flows of incompressible fluids in whichthe deviatoric part of the Cauchy stress and the symmetricpart of the velocity gradient are related through an implicitequation. Although we restrict ourselves to responses char-acterized by a maximal monotone graph, the structure isrich enough to include power-law type fluids, stress power-law fluids, Bingham and Herschel-Bulkley fluids, etc. Westudy Stokes-like problems wherein the inertial effects areneglected.
Agnieszka Swierczewska-GwiazdaUniversity of [email protected]
MS61
Non-Newtonian Fluids - Applications of Orlicz
Spaces in Theory of PDE
We are interested in the existence of solutions to stronglynonlinear partial differential equations, which come fromdynamics of non-Newtonian fluids of a nonstandard rhe-ology and abstract theory of parabolic equations. Thenonlinear highest order term is monotone and coerciv-ity/growth conditions are given by a general convex func-tion. We study both types of nonlinearity: sub- and super-linear. Such a formulation requires a general framework,therefore we work with non-reflexive and non-separable Or-licz and Musielak-Orlicz spaces.
Aneta Wroblewska-KaminskaInstitute of Mathematics [email protected]
MS62
Coupled Systems of Nonlinear Dispersive WaveEquations
Systems of wave equations which are coupled through non-linear and dispersive effects arise in a number of applica-tions. Unlike single equations, systems throw up far morediverse dynamics. This lecture will discuss a few of thesein the context of coupled Korteweg-de Vries systems.
Jerry BonaUniversity of Illinois at [email protected]
MS62
The Regularity Criteria for 3D Magneto-hydrodynamic System
This work studies the regularity problem for the 3DMagneto-Hydrodynamic (MHD) system. We apply an ap-proach of splitting the dissipation wavenumber combinedwith an estimate of the energy flux to obtain a new reg-ularity criterion. This splitting approach was originallyintroduced by Cheskidov and Shvydkoy. The regularitycriterion presented here is weaker than the existing criteriafor the 3D MHD system.
Alexey CheskidovUniversity of Illinois [email protected]
Mimi Dai, Mimi DaiUniversity of Colorado BoulderDepartment of Applied [email protected], [email protected]
MS62
The Atmospheric Equations of Water Vapor withSaturation
In this lecture we will recall the atmospheric equations ofwater vapor with saturation, appearing as a nonlinear mul-tivalued system of partial differential equations. We willaddress the issues of the definition of the solutions, andthe questions of existence, uniqueness and regularity of so-lutions.
Roger M. TemamInst. for Scientific Comput. and Appl. Math.Indiana [email protected]
110 PD13 Abstracts
Michele Coti ZelatiIndiana University, [email protected]
MS62
Averaging and Spectral Properties of the 2DAdvection-diffusion Equation in the Semi-classicalLimit for Vanishing Diffusivity
Abstract not available at time of publication.
Jesenko VukadinovicDepartment of MathematicsCollege of Staten Island, [email protected]
MS63
Relative Entropy in Hyperbolic Relaxation of Bal-ance Laws
We present a general framework for the approximation ofsystems of hyperbolic balance laws. The novelty of theanalysis lies on the construction of suitable relaxation sys-tems and the derivation of a relative entropy identity. Weprovide a direct proof of convergence in the smooth regimefor a wide class of physical systems. We present results forsystems arising in materials science, where the presence ofsource terms presents a number of additional challengesand requires delicate treatment. Our analysis is in thespirit and continuity of the previous work of A. Tzavaras(Comm. Math. Sci. 2005) for systems of hyperbolic con-servation laws.
Alexey MiroshnikovUniversity of Massachusetts [email protected]
Konstantina TrivisaUniversity of MarylandDepartment of [email protected]
MS63
On Multiphase Flow Models: Global Existence ofWeak Solutions
This work is part of a research program whose objectiveis the analysis of nonlinear systems via the constructionof suitable approximations and the establishment of ap-propriate compactness of the approximate solutions. Well-posedness results are presented for a class of such nonlinearsystems.
Konstantina TrivisaUniversity of MarylandDepartment of [email protected]
MS63
Global Weak Solutions to the InhomogeneousNavier-Stokes-Vlasov Equations
In this talk, we are going to talk about the global weaksolutions to the coupled system by the incompressible in-homogeneous Navier-Stokes equations and Vlasov equationin the three dimensional space.
Cheng Yu
University of [email protected]
Dehua WangUniversity of PittsburghDepartment of [email protected]
MS63
Shock-free Transonic Solutions to the Steady EulerSystem
We focus on the Ringleb exact solutions and build a familyof solutions nearby for the steady Euler system in two spacedimensions. For large amplitude flows, we implant a shockwhere the Ringleb-like solution would form a cusp.
Yuxi ZhengThe Pennsylvania State [email protected]
Tianyou ZhangThe Pennsylvania State Universityuniversity [email protected]
MS64
Long Time Influence of Small Perturbations
I will consider deterministic and stochastic perturbationsof dynamical systems and stochastic processes with mul-tiple invariant measures. Under certain conditions, theperturbed system in this case has a fast and a slow com-ponents. The slow component, which actually determinesthe long time behavior is a motion in the cone of invariantmeasures. The limiting slow motion can be described usingthe large deviation theory or by a modified averaging prin-ciple. I will demonstrate this approach in the case of theLandau-Lifshitz equation and for diffusion and wave frontsin narrow random channels.
Mark I. FreidlinUniversity of [email protected]
MS64
Asymptotic Behaviors for the Random SchrdingerEquation with Long-Range Correlations
Wave propagation in random media with long-range cor-relations was recently stimulated by data collections show-ing that propagation media can exhibit long-range correla-tion effects. Under the forward scattering approximation,the wave equation can be reduced to a random Schrdingerequation. In this talk, we will study the Schrdinger equa-tion with a slowly decorrelating random potential, andshow that the energy density of the wave function exhibitssome anomalous diffusion phenomena.
Christophe GomezAix-Marseille [email protected]
MS64
Linear Relaxation to Planar Travelling Waves in
PD13 Abstracts 111
Inertial Confinement Fusion
The models of ICF usually couple hydrodynamical featuresin plasmas with reaction-diffusion equations for the tem-perature. One of the main challenges is the non-linearheat diffusion of the porous media type, i-e ∇ · (λ∇T )with λ = λ(T ) = Tm−1 for some conductivity exponentm > 1. In this talk we will consider an approximate non-linear parabolic equation for which there exist planar wavesolutions having a physical meaning. Stability of thesewaves with respect to wrinkled perturbations is crucialfor the ICF process to be effective. We show here thata diffusion-induced mechanism forces relevant perturba-tions to become planar exponentially fast in the long timeregime, and also derive an asymptotic dispersion relationrelating this rate of relaxation to some small temperatureratio ε = Tmin/Tmax → 0. In this limit the problem de-generates into a free boundary one, and the proof involvesa rescaled singular principal eigenvalue problem.
Leonard MonsaingeonCarnegie Mellon [email protected]
MS64
Classical Limit for a System of Random Non-LinearSchrodinger Equations
This work is concerned with the semi-classical analysisof mixed state solutions to a Schrodinger-Poisson equa-tion perturbed by a random potential with weak ampli-tude and fast oscillations in time and space. We showthat the Wigner transform of the density matrix con-verges weakly and in probability to solutions to a Vlasov-Poisson-Boltzmann equation with a linear collision kernel.A consequence of this result is that a smooth non-linearitysuch as the Poisson potential (repulsive or attractive) doesnot change the statistical stability property of the Wignertransform observed in linear problems. We obtain in addi-tion that the local density and current are self-averaging,which is of importance for some imaging problems in ran-dom media. The proof brings together the martingalemethod for stochastic equations with compactness tech-niques for non-linear PDE in a semi-classical regime. Itpartly relies on the derivation of an energy estimate that isstraightforward in a deterministic setting but requires theuse of a martingale formulation and well-chosen perturbedtest functions in the random context.
Olivier PinaudColorado State [email protected]
MS65
High Frequency Homogenization: Connecting theMicrostructure to the Macroscale
It is highly desirable to be able to create continuum equa-tions that embed a known microstructure through effectiveor averaged quantities such as wavespeeds or shear moduli.The methodology for achieving this at low frequencies andfor waves long relative to a microstructure is well-knownand such static or quasi-static theories are well developed.However, at high frequencies the multiple scattering by theelements of the microstructure, which is now of a similarscale to the wavelength, requires a dynamic homogeniza-tion theory. Many interesting features of, say, periodicmedia: band gaps, localization etc occur at these higherfrequencies. The materials exhibit effective anisotropy and
this leads to topical effects such as cloaking/ invisibility,flat lensing, negative refraction and to inducing directionalbehaviour of the waves within a structure. A general the-ory will be described and applications to continuum, dis-crete and frame lattice structures will be outlined. Theresults and methodology are confirmed versus various illus-trative exact/ numerical calculations showing that theorycaptures, for instance, all angle negative refraction, ultrarefraction and localised defect modes.
Richard CrasterDepartment of MathematicsImperial College [email protected]
MS65
Quantum Graph Spectral Analysis of GraphyneStructures
We will discuss spectral features (band-gap structure,Dirac points, etc.) of various graphene and graphyne struc-tures. This is a joint work with N. T. Do and O. Post.
Peter KuchmentDepartment of Mathematics, Texas A&M [email protected]
MS65
Geometric Optimization of Laplace-BeltramiEigenvalues
Let (M, g) be a compact Riemannian surface and denote bylambdak(M, g) the k-th eigenvalue of the Laplace-Beltramioperator, Δg. In this talk, we consider the dependence ofthe mapping (M, g) → lambdak(M, g). In particular, wepropose a computational method for solving the eigenvalueoptimization problem of maximizing lambdak(M, g) over aconformal class [g] of fixed volume, vol(M, g) = 1. We alsoconsider the more general problem of letting M vary overall orientable compact surfaces of fixed genus. This is jointwork with Chiu-Yen Kao and Rongjie Lai.
Braxton OstingUniversity of California, Los [email protected]
MS65
Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials
As a first step toward a fully two-dimensional asymptotictheory for the bifurcation of solitons from infinitesimal con-tinuous waves, an analytical theory is presented for linesolitons, whose envelope varies only along one direction,in general two-dimensional periodic potentials. For thistwo-dimensional problem, it is no longer viable to rely ona certain recurrence relation for going beyond all ordersof the usual multiscale perturbation expansion, a key stepof the exponential asymptotics procedure previously usedfor solitons in one-dimensional problems. Instead, we pro-pose a more direct treatment which not only overcomesthe recurrence-relation limitation, but also simplifies theexponential asymptotics process. Using this modified tech-nique, we show that line solitons with any rational lineslopes bifurcate out from every Bloch-band edge; and foreach rational slope, two line-soliton families exist. Fur-thermore, line solitons can bifurcate from interior points ofBloch bands as well, but such line solitons exist only for
112 PD13 Abstracts
a couple of special line angles due to resonance with theBloch bands. In addition, we show that a countable setof multiline-soliton bound states can be constructed an-alytically. The analytical predictions are compared withnumerical results for both symmetric and asymmetric po-tentials, and good agreement is obtained. This is joint workwith Sean Nixon and T.R. Akylas.
Jianke YangDepartment of Mathematics and StatisticsUniversity of [email protected]
MS66
On the Temperature Variance Cascade in SurfaceQuasi-geostrophic Turbulence
Using a multi-scale, dynamical averaging methodology, werigorously establish the existence of a temperature variancecascade across a range of physical scales in surface quasi-geostrophic turbulence.
Zachary Bradshaw, Zoran GrujicUniversity of [email protected], [email protected]
MS66
On Dynamics of Fluid Flows in Porous Media
We study the degenerate parabolic equation with time-dependent flux boundary condition for generalized Forch-heimer (non-Darcy) flows of slightly compressible fluids inporous media. The solution is estimated, particularly forlarge time, in L∞-norm, W 1,r-norm for r ≥ 1, and W 2,2−δ-norm for δ > 0. The L∞-estimates of the solution’s timederivative are also obtained. We establish the continu-ous dependence of the solution on initial and boundarydata. The De Giorgi and Ladyzhenskaya-Uraltseva itera-tion techniques are combined with uniform Gronwall-typeestimates, specific monotonicity properties and suitableparabolic Sobolev embeddings. This is joint work withThinh Kieu and Tuoc Phan.
Luan HoangTexas Tech UniversityDepartment of Mathematics and [email protected]
MS66
On the Existence of Pullback Attractors for the 3DNavier-Stokes Equations
The existence of attractors for the 2D Navier-Stokes equa-tions has been thoroughly researched. In the nonau-tonomous case, when the uniform attractor exists, its sec-tions have attraction properties which are in a pullbacksense, when the initial conditions go to −∞. I will presentan abstract framework for studying the existence of pull-back attractors for the nonautonomous 3D Navier-Stokesequations, where uniqueness of solutions is yet unresolved.
Landon KavlieUniversity of Illinois at [email protected]
MS66
Rate of Vertical Heat Transport in Rayleigh-
Benard Convection
We discuss the scaling of the rate of heat transport in thevertical direction, quantified as the Nusselt number, withinthe Rayleigh-Benard model for convection. How the Nus-selt number scales in terms of the parameters of the system,Rayleigh number and Prandtl number, is an outstandingopen problem in classical physics. Both the case of finitePrandtl number, and the case of infinite Prandtl numberwill be discussed. Classical no-slip boundary condition aswell as geophysically relevant free-slip boundary conditionswill be investigated. The singular limit of infinite Prandtlnumber limit will also be discussed both in terms of finite-time trajectory convergence and in terms of convergence oflong time statistical properties.
Xiaoming [email protected]
MS67
Two Dimensional Water Waves In HolomorphicCoordinates
This article is concerned with the in nite bottom waterwave equation in two space dimensions. We consider thisproblem expressed in position-velocity potential holomor-phic coordinates. Viewing this problem as a quasilineardispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solu-tions for small localized data. Neither of these results arenew; they have been recently obtained by Alazard-Burq-Zuily, respectively by Wu using different coordinates andmethods. Instead our goal is improve the understanding ofthis problem by providing a single setting for both results,as well as new, somewhat simpler proofs.
Mihaela IfrimMcMaster [email protected]
Daniel TataruUniversity of California, [email protected]
John HunterUniversity of California, [email protected]
MS67
On the Spectral Stability of Kinks in Some PT-symmetric Nonlinear Equations
We consider the introduction of PT-symmetric terms inthe context of classical Klein-Gordon field theories and ex-plore their implications on the spectral stability of coher-ent structures. These models take the form utt − uxx +εγ(x)ut+ f(u) = 0, where γ(x) is an odd function. In par-ticular, for the corresponding static kink solutions of thesine-Gordon equation and the Klein-Gordon equation withnonlinearity f(u) = 2(u3 − u) we prove spectral stabilityfor all small enough ε.
Milena StanislavovaUniversity of Kansas, LawrenceDepartment of Mathematics
PD13 Abstracts 113
MS67
Global Exact Controllability of Semilinear PlateEquations
I will discuss some exact controllability results for semi-linear hyperbolic PDE’s. In early 1990’s it was demon-strated by Lasiecka and Triggiani that the global exactinterior and boundary controllability holds for many waveand plate models with nonlinear source terms, providedthe source feedback maps are globally Lipschitz. In thespecial case of localized interior controls these theoremshave been extended by Li and Zhang to sources that areat most ”logarithmically superlinear” at infinity. I willpresent some nascent developments concerning global ex-act controllability for plate equations with more generalpolynomially bounded sources.
Matthias EllerGeorgetwon UniversityDepartment of [email protected]
Daniel ToundykovUniversity of [email protected]
MS68
Upscaling of Forchheimer Flows
In this work we propose upscaling method for nonlin-ear Forchheimer flow in highly heterogeneous porous me-dia. The generalized Forchheimer law is considered for in-compressible and slightly-compressible single-phase flows.We use recently developed analytical results by E.Aulisa,L.Bloshaskaya, L.Hoang, and A.Ibragimov [J. Math. Phys.50, 103102 (2009)] and formulate the resulting system interms of a degenerate nonlinear flow equation for the pres-sure with the nonlinearity depending on the pressure gradi-ent. The coarse scale parameters for the steady state prob-lem are determined so that the volumetric average of ve-locity of the flow in the domain on fine scale and on coarsescale are close. A flow-based coarsening approach is used,where the equivalent permeability tensor is first evaluatedfollowing streamline methods for linear cases, and modifiedin order to take into account the nonlinear effects. Thedeveloped upscaling algorithm for nonlinear steady stateproblems is effectively used for variety of heterogeneitiesin the domain of computation. Direct numerical computa-tions for average velocity and productivity index justify theusage of the coarse scale parameters obtained for the spe-cial steady state case in the fully transient problem. Fornonlinear case analytical upscaling formulas in stratifieddomain are obtained. Numerical results were compared tothese analytical formulas and proved to be highly accu-rate. This is the joint work with E.Aulisa, A.Ibragimovand Y.Efendiev.
Lidia BloshanskayaSUNY at New Paltz, Department of [email protected]
Eugenio AulisaDepartment of Mathematics and Statistics.Texas Tech [email protected]
Akif Ibragimov
Texas [email protected]
Yalchin EfendievDept of MathematicsTexas A&M [email protected]
MS68
Two-phase Generalized Forchheimer Flows
We study two-phase generalized Forchheimer flows of in-compressible fluids in porous media. A family of non-constant symmetric steady states are obtained. We provethat they are linearly asymptotically stable in case ofbounded domains. For unbounded domains, we obtain thestability results and the behavior of the solutions of the lin-earized system at space-infinity. The lemmas of growth ofLandis-type are established with explicit barrier functionsusing the particular structure of the steady states. This isjoint work with Luan Hoang and Akif Ibragimov.
Thinh KieuTexas Tech [email protected]
Luan HoangTexas Tech UniversityDepartment of Mathematics and [email protected]
Akif IbragimovTexas [email protected]
MS68
Complementary Media in Metamaterials
In this talk, I will discuss the frame work of complementarymedia in metamaterials. Resonant and localized resonantphenomena will be discussed. Applications in cloaking willbe given.
Hoai-Minh NguyenUniversity of [email protected]
MS68
On Passage to the Limit in Nonlinear Elliptic andParabolic Equations
Passing to the limit in nonlinear terms one encounters thefundamental problem of ”weak convergence of fluxes toa flux”. The term ”flux” refers to the vector A(x,∇uε)which stands in the elliptic equation under the divergencesign. Usually one has the weak convergence uε ⇀ u insome Sobolev space and the weak convergence of the fluxesA(x,∇uε) ⇀ z in some Lebesgue space. This leads to theproblem of the equality z = A(x,∇u). Let us state one re-sult in this direction. Assume that 1. uε ⇀ u in W 1,α(Ω),α > 1; 2. The symbol A is monotone:
(A(x, ξ)−A(x, η)) · (ξ − η) ≥ 0;
3. divA(x,∇uε) = 0 in the sense of distributions,
Aε ≡ A(x,∇uε) ⇀ z in Lβ′(Ω), β′ =
β
β − 1;
114 PD13 Abstracts
4. 1 < α ≤ β < α∗,
α∗ =
{α(N−1)N−1−α
for α < N − 1,
+∞ for α ≥ N − 1;
5. The family of ”energy densities” Aε · ∇uε is boundedin L1(Ω); 6. uε ∈ W 1,β(Ω) (additional regularity of pre-limit functions). Then z = A(x,∇u). (Note that in theclassical case when α = β the last three conditions holdautomatically.) In this talk we will discuss the followingapplications: monotone elliptic operators with nonstan-dard growth conditions; thermistor problem; and gener-alized Navier-Stokes equations.
Vasily ZhikovVladimir Pedagogical State UniversityVladimir, [email protected]
MS69
A Blob Method for the Aggregation Equation
In the past five to ten years, there has been substantial ac-tivity on the theory of the aggregation equation, but com-paratively little on the analysis of numerical methods. Inthis talk, I will present joint work with Andrea Bertozzi ona blob method for the aggregation equation, inspired byvortex blob methods from the classical fluids case.
Katy CraigRutgers [email protected]
MS69
Nonlinear Stability of Steady States of Nonlocal-interaction Equations
We will discuss stability of steady states of the nonlocal-interaction equations. The problem is considered in thespace of measures. Numerical experiments show that sta-ble steady states of nonlocal interaction equations can ex-hibit a wide array of patterns. Recent works also show thatfor smooth potentials the energy minimizers are supportedon set of dimension less than one. We will discuss rigorousresults on the stability of steady states for smooth poten-tials. The main focus is on (local) minimizers which aresums of delta masses. We establish sufficient conditions forthe nonlinear stability to hold. The criteria are based onextended notion of linear stability, which allows for split-ting of the particles, and can be verified for a number ofsteady states.
Dejan Slepcev, Robert SimioneCarnegie Mellon [email protected], [email protected]
MS69
Title Not Available at Time of Publication
Abstract not available at time of publication.
David T. UminskyUniversity of San FranciscoDepartment of [email protected]
MS69
Nonlocal Interaction Equations in Environments
with Heterogeneities and Boundaries
In this presentation, we discuss biological aggregation in aheterogeneous environment by modeling the environmentas a Riemannian manifold with boundary. We develop gra-dient flow approach in the space of probability measureson the manifold endowed with Riemannian 2-Wassersteinmetric. We then show the existence and stability of weakmeasure solutions to a class of nonlocal interaction equa-tions through the gradient flow approach. We discuss howheterogeneity of the environment leads to new dynamicalphenomena. In particular exposed to a unidirectional ex-ternal potential the agents form a rolling travelling wave(in contrast to the translational motion in the homoge-neous environment). We close the presentation by givingsome simulations to show the rolling phenomenon.
Lijiang Wu, Dejan SlepcevCarnegie Mellon [email protected], [email protected]
MS70
Liquid Crystals: from Molecular Dynamics to Pdes
Abstract not available at time of publication.
Ibrahim FatkullinUniversity of ArizonaDepartment of [email protected]
MS70
Field Responses of Smectic A Liquid Crystals in2D and 3D
We study the Landau-de Gennes free energy to describethe undulatory instability in smectic A liquid crystals sub-jected to magnetic fields. At the onset of undulations closeto the critical field, the periodic oscillation occurs. Weprove this phenomena by the bifurcation theory to the non-linear system of Landau-de Gennes model. We also find thecritical field and oscillatory description at the onset of theundulations. When the applied field is sufficiently large,the director lies in the direction either positive or nega-tive field assuming the natural boundary condition on thesmectic order parameter is imposed. We also perform nu-merical simulations to illustrate the results of our analysis.Undulation instabilities on three dimensional systems willbe also discussed.
Sookyung JooOld Dominion [email protected]
Carlos Garcia-CerveraMathematics, [email protected]
MS70
The Existence of Global Solutions to the Ericksen-Leslie System in R2
In this talk, I will discuss the regularity and existence ofglobal weak solution of the general Ericksen-Leslie systemin dimension two under a set of condition of Leslie coef-ficients, which assure that the total energy is decreasingunder the flow. This is based on joint work with Jinrui
PD13 Abstracts 115
Huang and Fanghua Lin.
Changyou Wangdepartment of mathematicsuniversity of [email protected]
MS70
Models for Active Liquid Crystals and Their Ap-plications to Complex Biological Systems
Abstract not available at time of publication.
Qi WangUniversity of South [email protected]
MS71
The Wiener Test for the Removability of the Loga-rithmic Singularity for the Elliptic PDEs with Mea-surable Coefficients, and Its Measure-Theoretical,Topological, and Probabilistic Consequences
This talk introduces a notion of log-regularity (or log-irregularity) of the boundary point ζ (possibly ζ = ∞) ofarbitrary open subset Ω of the Greenian deleted neigbor-hood of ζ in R2 concerning second order uniformly ellipticequations with bounded and measurable coefficients, ac-cording as whether the log-harmonic measure of ζ is null(or positive). A necessary and sufficient condition for theremovability of the logarithmic singularity, that is to sayfor existence of a unique solution to the Dirichlet problemin Ω in a class of solutions with possibly logarithmic growthrate at ζ, is established in terms of the Wiener test for thelog-regularity of ζ. From the topological point of view, theWiener test at ζ presents minimal thinness criteria of setsnear ζ in minimal fine topology. Precisely, the open set Ωis a deleted neigbourhood of ζ in minimal fine topology ifand only if ζ is log-irregular. From the probabilistic pointof view Wiener test presents asymptotic law for the log-Brownian motion near ζ conditioned on the logarithmickernel with pole at ζ.
Ugur G. AbdullaDepartment of Mathematical SciencesFlorida Institute of [email protected]
MS71
The Elliptic Obstacle Problem When the Coeffi-cients Are Only in Vmo
We study the elliptic obstacle problem in both divergenceand nondivergence form. In both cases, although we canshow a measure theoretic version of Caffarelli’s Theoremfrom his famous 1977 Acta paper, we can also give exam-ples of nonuniqueness in blowup limits.
Ivan Blank, Zheng HaoKansas State [email protected], [email protected]
Kubrom TekaSUNY Oswego
MS71
Monotonicity Formulas in the Signorini Problem
We will discuss some new monotonicity formulas whichplay a crucial role in the study of the Signorini problem.
Nicola GarofaloUniversita’ di [email protected]
MS71
On a Thermodynamically Consistent Stefan Prob-lem with Variable Surface
A thermodynamically consistent two-phase Stefan prob-lem with temperature-dependent surface tension and withor without kinetic undercooling is studied. It is shownthat these problems generate local semiflows in well-definedstate manifolds. If a solution does not exhibit singularities,it is proved that it exists globally in time and converges to-wards an equilibrium of the problem. In addition, stabilityand instability of equilibria is studied. In particular, itis shown that multiple spheres of the same radius are un-stable if surface heat capacity is small; however, if kineticundercooling is absent, they are stable if surface heat ca-pacity is sufficiently large. (Joint work with J. Pruss andM. Wilke).
Gieri SimonettVanderbilt [email protected]
MS72
Modeling Arterial Walls As a Multi-Layered Struc-ture, and Their Interaction with Pulsatile BloodFlow
We study the interaction between an incompressible, vis-cous fluid and a multilayered structure, which consists ofa thin and a thick elastic layer. The thin layer is modeledusing the linearly elastic Koiter shell model, while the thicklayer is modeled using the 2D equations of linear elastic-ity. The system is coupled via the kinematic and dynamiccoupling conditions. We present a loosely-coupled numer-ical algorithm, and stability results. The results will besupported with numerical examples.
Martina BukacUniversity of PittsburghDepartment of [email protected]
Boris MuhaDepartment of Mathematics, Faculty of ScienceUniversity of [email protected]
Suncica CanicDepartment of MathematicsUniversity of [email protected]
MS72
Validation of An Open Source Framework for the
116 PD13 Abstracts
Simulation of Blood Flow in Deformable Vessels
We discuss in the validation of an open source frame-work for the solution of problems arising in hemodynamics.The framework is assessed through a numerical benchmarkdealing with the propagation of a pressure wave in a fluid-filled elastic tube. The computed pressure wave speed andfrequency of oscillations, and the axial velocity of the fluidon the tube axis are close to the values predicted by theanalytical solution associated with the benchmark.
Tiziano PasseriniDepartment of Math & CSEmory [email protected]
Annalisa QuainiDepartment of Mathematics, University of [email protected]
Umberto E. VillaDept. of Mathematics and Computer ScienceEmory [email protected]
Alessandro VenezianiMathCS, Emory University, Atlanta, [email protected]
Suncica CanicDepartment of MathematicsUniversity of [email protected]
MS72
Reduced Models for Solving Inverse Fluid-Structure Interaction Problems
We are interested in solving the data assimilation prob-lem: given the displacement of a pipe where an incom-pressible fluid flows, identify the wall compliance (Youngmodulus E). We resort to a variational approach. E is re-garded as control variable in the minimization of the mis-match between measured and computed displacement. Insolving this inverse fluid-structure interaction problem, weface formidable computational costs. We resort to reducedmodels based on a Proper Orthogonal Decomposition ap-proach.
Luca BertagnaDept. Math & CSEmory [email protected]
Alessandro VenezianiMathCS, Emory University, Atlanta, [email protected]
MS72
Enforcing Interface Conditions for Fsi ProblemsUsing Nitsche’s Method. Derivation of ExplicitCoupling Strategies for Multilayered PoroelasticArteries.
Nitsche’s method was developed to weakly enforce Dirichletboundary conditions for PDEs approximated with FEM.Recently, it has been applied to the transmission conditionsof fluid-structure interaction (FSI) problems. It provides a
general framework capable to accommodate strongly andloosely coupled schemes. We apply it to design FSI algo-rithms for multilayered poroelastic arteries. We will com-bine the analysis of the equations with direct simulationsto understand how poroelasticity affects blood flow in ar-teries.
Martina BukacUniversity of PittsburghDepartment of [email protected]
Ivan YotovUniveristy of PittsburghDepartment of [email protected]
Rana ZakerzadehDepartment of Mechanical Engineering and MaterialsScienceUniversity of [email protected]
Paolo ZuninoUniversity of PittsburghDepartment of [email protected]
MS73
Singular Limits of Compressible Fluids
We discuss several singular limits arising in the scale anal-ysis of the Navier-Stokes(-Fourier) system describing themotion of a general compressible, viscous, (and heat con-ducting) fluid. We consider rapidly rotating stratified flu-ids in the incompressible and/or inviscid limit. The effectof simultaneous scalings is considered. To this end, we usethe concept of relative entropy, together with careful anal-ysis of certain oscillatory integrals arising in the study ofPoincare waves.
Eduard FeireislMathematical Institute ASCR, Zitna 25, 115 67 Praha 1Czech [email protected]
MS73
Compressible Navier-Stokes System with InflowCondition
I will talk about the issue of stability of Poiseuille typeflows in regime of compressible Navier-Stokes equationsin a three dimensional finite pipe-like domain. We provethe existence of stationary solutions with inhomogeneousNavier slip boundary conditions admitting nontrivial in-flow condition in the vicinity of constructed generic flows.Our techniques are based on an application of a modifi-cation of the Lagrangian coordinates. Thanks to such ap-proach we are able to overcome difficulties coming from hy-perbolicity of the continuity equation, constructing a max-imal regularity estimate for a linearized system and apply-ing the Banach fixed point theorem. The talk is based onthe joint result with Tomasz Piasecki.
Piotr MuchaWarsaw UniversityMathematics Department
PD13 Abstracts 117
MS73
Well-Posedness of the Primitive Equations of theOcean with Continuous Initial Data
We address the well-posedness of the primitive equations ofthe ocean with continuous initial data. We show that thesplitting of the initial data into a regular finite energy partand a small bounded part is preserved by the equationsthus leading to existence and uniqueness of solutions.
Walter RusinUniversity of Southern [email protected]
Walter RusinOklahoma State [email protected]
Igor Kukavica, Yuan Pei, Mohammed ZianeUniversity of Southern [email protected], [email protected], [email protected]
MS73
Existence Analysis of the Navier-Stokes System forMulticomponent Mixtures
We will present a model of motion of compressible, heat-conducting mixture of arbitrary large number of species.We consider a special form of density-dependent viscositycoefficients and a singular behaviour of the cold compo-nent of the internal pressure near vacuum. Under thesehypotheses we prove global-in-time existence of weak so-lutions. This result is based on several joint papers withP.B. Mucha and M. Pokorny.
Ewelina ZatorskaInstitute of Applied Mathematics and MechanicsWarsaw University, [email protected]
MS74
Pde Model of Gbm: Insights Into The Shortcom-ings of Anti-Angiogeneic Therapy
The recent use of anti-angiogenesis (AA) drugs for thetreatment of glioblastoma multiforme (GBM) has uncov-ered unusual tumor responses. Here, we derive a newmathematical model that takes into account the ability ofproliferative cells to become invasive under hypoxic condi-tions; model simulations generate the multilayer structureof GBM, namely proliferation, brain invasion, and necro-sis. The model is validated by replicating and justifyingthe clinical observation of rebound growth when AA ther-apy is discontinued in some patients. The model is inter-rogated to derive fundamental insights in cancer biologyand on the clinical and biological effects of AA drugs. In-vasive cells promote tumor growth, which in the long-runexceeds the effects of angiogenesis alone. Furthermore, AAdrugs increase the fraction of invasive cells in the tumor,which explain progression by FLAIR and the rebound tu-mor growth when AA is discontinued.
Hassan M. Fathallah-ShaykhUniversity of Alabama at [email protected]
Olivier SautInstitut de Mathematiques de Bordeaux, [email protected]
Thierry ColinInstitut de Mathematiques de BordeauxInstitut Polytechnique de [email protected]
MS74
On the Euler-Poincare Equation with Non-ZeroDispersion
The Euler-Poincare equation was introduced by Holm,Marsden and Ratiu to study ideal fluids with nonlinear dis-persion. It can be viewed as a natural multi-dimensionalgeneralization of the popular Camassa-Holm equations. Iwill discuss some recent results on the sharp characteri-zation of a special family of solutions based on some newinverse-Soblev type inequalities. In particular this settlesan open problem raised by Chae and Liu.
Dong LiDepartment of MathematicsUniversity of British [email protected]
Xinwei Yu, Zhichun ZhaiUniversity of [email protected], [email protected]
MS74
Laminar Boundary Layers in Rayleigh-BenardConvection
We study Rayleigh-Benard convection in the high-Rayleigh-number and infinite-Prandtl-number regime.While the dynamics in the bulk are characterized by achaotic convective heat flow, the boundary layers at thehorizontal container plates are essentially conducting andthus the fluid is motionless. Consequently, the averagetemperature exhibits a linear profile in the boundary lay-ers. In this talk, I present new rigorous local bounds ontemperature field and fluid velocity in the boundary layers,and show that the temperature profile is indeed essentiallylinear. All results are uniform in the system parameters(e.g. the Rayleigh number) up to logarithmic correctionterms.
Christian SeisUniversity of [email protected]
MS75
Weak Solutions for Pressureless Gas Dynamics andOther Hyperbolic Systems Through Energy Mini-mization
We introduce a new and very simple method to obtainweak solutions to some hyperbolic problems, including gasdynamics in dimension 1 (any gamma between 1 and 3) andpressureless gas dynamics in any dimension. The methoditself simply consists in minimizing the kinetic energy overthe considered time interval with appropriate constraints.
Pierre-Emmanuel JabinUniversity of Maryland
118 PD13 Abstracts
Jean-Francois JabirUniversity of Santiago, [email protected]
Alexis F. VasseurUniversity of Texas at [email protected]
Amina AmassadUniversity of [email protected]
MS75
Title Not Available at Time of Publication
Abstract not available at time of publication.
Alexis F. VasseurUniversity of Texas at [email protected]
MS75
Incompressible Limits for Magnetohydrodynamics
Incompressible limits for magnetohydrodynamics will bediscussed. In particular, the zero Mach limit and the hy-drodynamic limit will be presented.
Dehua WangUniversity of PittsburghDepartment of [email protected]
MS75
Convex Entropy, Hopf Bifurcation, and Viscousand Inviscid Shock Stability
We discuss relations between one-dimensional inviscid andviscous stability/bifurcation of shock waves in continuum-mechanical systems and existence of a convex entropy. Inparticular, we show that the equations of gas dynamicsadmit equations of state satisfying all of the usual assump-tions of an ideal gas, along with thermodynamic stability-i.e., existence of a convex entropy- yet for which there oc-cur unstable inviscid shock waves. For general 3x3 systems(but not up to now gas dynamics), we give numerical evi-dence showing that viscous shocks can exhibit Hopf bifur-cation to pulsating shock solutions. Our analysis of inviscidstability in part builds on the analysis of R. Smith charac-terizing uniqueness of gas dynamical Riemann solutions interms of the equation of state of the gas, giving an analo-gous criterion for stability of individual shocks.
Kevin ZumbrunIndiana [email protected]
MS76
Coherent Structures of a Nonlocal FitzHugh-Nagumo Equation
We prove the existence of coherent structures for a class ofFitzHugh-Nagumo equations with nonlocal diffusion. Un-like the dynamical systems approach via geometric singu-
lar perturbation theory (Fenichel’s theorem and ExchangeLemma), our proof relies on matched asymptotics tech-niques and Fredholm properties of differential operatorson suitable Banach spaces (Spectral Flow and NonlocalExchange Lemma).
Gregory FayeUniversity of Minnesota, [email protected]
Arnd ScheelUniversity of [email protected]
MS77
Deterministic Solvers to Describe DG-MOSFETDevices
Statistical models are used to describe electron transportin semiconductors at a mesoscopic level. The basic modelis given by the Boltzmann transport equation for semicon-ductors in the semi-classical approximation coupled withPoissons equation, since the electric ?eld is self-consistentdue to the electrostatics produced by the electrons and thedopants in the semiconductor. However, for short devices,this mathematical description has to be modified, sincethe electrons behave as particles along the long dimension,and as waves along the short dimension. In these cases,the devices can be modelled by the Boltzmann-Schrdinger-Poisson system. In this talk we show deterministic solversto describe long and short Double Gate Metal Oxide Semi-conductor Field E?ect Transistors (DG-MOSFET) devices.
Maria CaceresUniversidad de Granada, [email protected]
MS77
Charge Transport in Bulk-Heterojunction OrganicSolar Cells: Adaptivity, High-Throughput Com-puting and Optimal Morphologies
Abstract not available at time of publication.
Baskar GanapathysubramanianIowa State [email protected]
MS77
Multiscale Modeling and Computation of NanoOptical Responses
We introduce a new framework for the multiphysicalmodeling and multiscale computation of nano-optical re-sponses. The semi-classical theory treats the evolution ofthe electromagnetic field and the motion of the chargedparticles self-consistently by coupling Maxwell equationswith Quantum Mechanics. To overcome the numer-ical challenge of solving high dimensional many bodySchrodinger equations involved, we adopt the Time Depen-dent Current Density Functional Theory (TD-CDFT). Inthe regime of linear responses, this leads to a linear systemof equations determining the electromagnetic eld as wellas the current and electron densities simultaneously. Aself-consistent multiscale method is proposed to deal withthe well separated space scales. Numerical examples are
PD13 Abstracts 119
presented to illustrate the resonant condition.
Di LiuMichigan State UniversityDepartment of [email protected]
MS77
First-Principles Prediction of Charge and EnergyTransport in Disordered Semiconductors for Pho-tovoltaic Applications
Abstract not available at time of publication.
Gang LuCalifornia State University [email protected]
MS78
Homogenization for Problems with OscillatoryBoundary Conditions
In this talk I will discuss two new homogenization results,one concerning a growing free boundary in a randomlyperforated domain (with Inwon Kim) and another deal-ing with a fully nonlinear Neumann problem with periodicdata (with Russell Schwab). In both instances, the mainresult is that the oscillations of the boundary data or ofthe random perforations resolve themselves in such a waythat the problems converge in the high frequency limit toa deterministic and homogeneous problem. Although bothproblems are of a very different nature, their broad anal-ysis centers on two key themes: the derivation of uniformpointwise estimates and the identification of an effectiveequation via auxiliary correctors. In the case of the fullynonlinear Neumann problem, the effective equation at theboundary can be identified thanks to a new representationformula for nonlinear operators enjoying a global compar-ison principle.
Nestor GuillenDepartment of Mathematics [email protected]
MS78
Anomalous Spreading in a System of CoupledFisher-Kpp Equations
This talk will concern anomalous spreading in a system ofreaction-diffusion equations. When a single parameter isset to zero the system consists of a pair of uncoupled Fisher-KPP equations. Anomalous spreading occurs when onecomponent of the system spreads at a faster speed in thecoupled system than it does in isolation, while the speed ofthe second component remains unchanged. We will discussthe role of the pointwise Green’s function in anomalousspreading and discuss mechanisms leading to anomalousspreading in both the linear and nonlinear regime.
Matt HolzerSchool of MathematicsUniversity of [email protected]
MS78
The Logarithmic Delay of Kpp Fronts in a Hetero-
geneous Medium
I’ll describe the asymptotic behavior of solutions to areaction-diffusion equation with KPP-type nonlinearity.One can interpret the solution in terms of a branchingBrownian motion. It is well-known that solutions to theCauchy problem may behave asymptotically like a travelingwave moving with constant speed. On the other hand, forcertain initial data, M. Bramson proved that the solutionto the Cauchy problem may lag behind the traveling waveby an amount that grows logarithmically in time. UsingPDE arguments, we have extended this statement aboutthe logarithmic delay to the case of periodically varyingreaction rate. We also consider situations where the de-lay is larger than logarithmic. This new PDE approachinvolves the study of the linearized equation with Dirichletcondition on a moving boundary.
Francois HamelUniversite d’[email protected]
James NolenDuke UniversityMathematics [email protected]
Jean-Michel RoquejoffreUniversite Paul [email protected]
Lenya RyzhikStanford [email protected]
MS78
Sharp Asymptotic Growth Laws of TurbulentFlame Speeds in Cellular Flows by InviscidHamilton-Jacobi Models
To determine the turbulent flame speed is a fundamentalproblem in turbulent combustion. A specific project is tounderstand its growth law with respect to the flow inten-sity. In this talk, we will present the sharp growth lawof turbulent flame speed in two dimensional cellular flowspredicted by a model proposed by Majda and Souganidisin 90’s. This answers an open problem posed in a paperby Embid, Majda and Souganidis.
Yifeng YuUniversity of [email protected]
Jack XinDepartment of [email protected]
MS79
Landis-type Lemma of Growth in the LayeredCylinders for Solution of the Parabolic DegenerateEquation and Applications
We will consider a parabolic equation of second order in un-bounded domain with Dirichlet type boundary conditions.Corresponding operator can degenerate at infinity with re-spect to x and t. Lemma of growth will be proved for classof the solution of the equation in layered cylinder. Using
120 PD13 Abstracts
this Lemma we will prove parabolic analog Phragmen-Lindelof type principle in x and t directions.
Akif IbragimovTexas [email protected]
Luan HoangTexas Tech UniversityDepartment of Mathematics and [email protected]
Thinh KieuTexas Tech [email protected]
MS79
An Ersatz Existence Theorem for Fully NonlinearParabolic Equations without Convexity Assump-tions
We show that for any uniformly parabolic fully nonlinearsecond-order equation with bounded measurable “coeffi-cients’ and bounded “free’ term in the whole space or inany cylindrical smooth domain with smooth boundary dataone can find an approximating equation which has a contin-uous solution with the first and the second spatial deriva-tives under control: bounded in the case of the whole spaceand locally bounded in case of equations in cylinders. Theapproximating equation is constructed in such a way thatit modifies the original one only for large values of the sec-ond spatial derivatives of the unknown function. This isdifferent from a previous work of Hongjie Dong and theauthor where the modification was done for large values ofthe unknown function and its spatial derivatives.
Nicolai KrylovUniversity of [email protected]
MS79
Ultrasound-modulated Optical Tomography: ANonlinear Model
Let Ω be an optical body with several obstacles and leta∗ denote the unknown absorption coefficient of Ω. Suchobstacles can be imaged using the knowledge of a∗. Inthis talk, by using near-infrared light to illuminate Ω andspherical acoustic waves to purturb Ω, we can establish anonlinear equation for a∗. This equation can be solved byapplying the optimal control method.
Loc NguyenEcole Normale Superrieure, [email protected]
MS79
A New Curvature-dependent Surface TensionModel in Fracture Mechanics
A new approach to modeling of a brittle fracture based onan extension of a continuum mechanics to the nanoscalewill be considered. The classical Neumann boundarycondition on the fracture surface is augmented with acurvature-dependent surface tension. The model is stud-ied on an example of a plain-strain mixed mode fracture.Using complex variable methods, the mechanical problemis reduced to a system of coupled Cauchy singular integro-
differential equations. The regularization and the numeri-cal solution of this system is considered. It is shown that in-corporation of a curvature- dependent surface tension intofracture modeling eliminates the integrable crack-tip stressand strain singularities of order 1/2 present in the classi-cal linear fracture mechanics solutions. Directions for thefuture research in this area will be presented.
Anna ZemlyanovaDepartment of MathematicsTexas A&M [email protected]
MS80
Global Regularity for Some Oldroyd-B Type Mod-els of Viscoelasticity
We investigate some critical models for visco-elastic flowsof Oldroyd-B type in dimension two. We use a transfor-mation which exploits the Oldroyd-B structure to provean L∞ bound on the vorticity which allows us to proveglobal regularity for our systems. This is a joint work withFrederic Rousset
Tarek ElghindiCourant [email protected]
MS80
Title Not Available at Time of Publication
Abstract not available at time of publication.
Taoufik HmidiUniversity of Rennes [email protected]
MS80
The Numerical Approach of the Singularly Per-turbed Problems
In this talk, I will present the numerical solutions of sin-gularly perturbed problems in a circular domain and pro-vide as well approximation schemes, error estimates andnumerical simulations. To resolve the oscillations of classi-cal numerical solutions due to the stiffness of our problem,we construct, via boundary layer analysis, the so-calledboundary layer elements which absorb the boundary layersingularities. Using a P1 classical finite element space en-riched with the boundary layer elements, we obtain an ac-curate numerical scheme in a quasi-uniform mesh. The talkincludes a joint work with C.-Y. Jung and R. Temam.
YoungJoon HongIndiana [email protected]
MS80
Title Not Available at Time of Publication
Abstract not available at time of publication.
Slim IbrahimUniversity of Victoria
PD13 Abstracts 121
MS81
Magic Toplitz Matrix for Mixed Boundary LaplaceEquation and Its Application to Breast Cancer De-tection
Laplace equation naturally arises in modeling of electricalproperties of the media and can be used to recover conduc-tivity and permittivity inside the body through measure-ment of currents and voltages on the periphery of the body.In electrical impedance tomography (EIT), currents are in-jected through the electrodes located on the boundary ofthe body with no current between electrodes which leads toa mixed boundary value problem. The inverse problem onthe reconstruction of conductivity and permittivity at eachpoint inside the body is multidimensional and ill-posed.We suggest that for cancer detection/screening one needsto prove that the breast tissue is non-homogeneous. Thisleads to the problem of derivation of the exact solution ofthe Laplace equation with mixed-type boundary conditions(Robin-type on the electrodes and Neumann-type betweenelectrodes). Under simplifying assumption, this solutionis derived for the disk that gives rise to Magic Toeplitz(circulant) matrix which can be viewed as a Dirichlet-to-Neumann map, or in the context of physics as a general-ization of the Ohms law on the disk. It is shown that theelements of the Magic Toeplitz matrix can be representedthrough new special function that is expressed via Fourierintegral or series. We apply Magic Toeplitz matrix to de-tect bad electrode contact on the breasta typical problemof EIT in a clinical setting.
Eugene DemidenkoDarthmouth [email protected]
MS81
On the Optimal Control of Free Boundary Bound-ary Problems for the Second Order Parabolic PDEs
We develop a new variational formulation of the inverseStefan problem, where information on the heat flux on thefixed boundary is missing and must be found along with thetemperature and free boundary. We employ optimal con-trol framework, where boundary heat flux and free bound-ary are components of the control vector, and optimalitycriteria consists of the minimization of the sum of L2-normdeclinations from the available measurement of the temper-ature flux on the fixed boundary and available informationon the phase transition temperature on the free bound-ary. This approach allows one to tackle situations whenthe phase transition temperature is not known explicitly,and is available through measurement with possible error.It also allows for the development of iterative numericalmethods of least computational cost due to the fact thatfor every given control vector, the parabolic PDE is solvedin a fixed region instead of full free boundary problem.We prove well-posedness in Sobolev spaces framework andconvergence of discrete optimal control problems to theoriginal problem both with respect to cost functional andcontrol.
Ugur G. AbdullaDepartment of Mathematical SciencesFlorida Institute of [email protected]
Jonathan Goldfarb
Florida Institute of [email protected]
MS81
Recent Progress on Local Behavior of the Logarith-mic Diffusion Equation
This talk describes some recent progress on the local behav-ior of the equation ut = Δ ln u including a Harnack-typeinequality, L1 form Harnack inequality and local specialanalyticity. Connection with the porous medium equationut = Δ(um/m) and stability of all the estimates as m tendsto 0 is presented.
Naian LiaoVanderbilt [email protected]
Emmanuele DiBenedettoDepartment of MathematicsVanderbilt University, [email protected]
Ugo P. GianazzaDepartment of MathematicsUniversity of [email protected]
MS81
Title Not Available at Time of Publication
Abstract not available at time of publication.
Jean-Paul ZolesioCNRS and INRIASophia Antipolis, [email protected]
MS82
Kinetic Formulation and Regularizing Effects forDegenerate Parabolic Equations
In this talk, we present how to obtain regularity of solutionsto degenerate parabolic equations. The main idea is toidentify the renormalization property of solutions from thestructure of the nonlinearity. We then apply the velocityaveraging to their underlying kinetic formulations, whichleads to the regularizing effect of solutions. Our resultsextend previous regularity statements and provide a newregularity result of renormalized solutions.
Hantaek BaeUniversity of California, [email protected]
Eitan TadmorUniversity of [email protected]
MS82
Critical Regularity Estimates for Transport Prob-lems
A new method to obtain regularity for solutions to trans-port equations is introduced. This is applied to problems inbiology (such as some chemotaxis models) but also to com-
122 PD13 Abstracts
pressible fluid dynamics where it significantly extends theexisting existence theory (non monotone pressure terms,temperature dependent viscosity... In addition of provid-ing compactness for the macroscopic density, the methodalso yields explicit, critical regularity estimates.
Pierre-Emmanuel JabinUniversity of [email protected]
MS82
Regularizing Effects on 1D Nonlinear Scalar Con-servation Laws in Fractional Bv Spaces
In 1D, fractional BV spaces are introduced to obtainsmoothing effects for conservation laws. Theses spaces havetrace properties like BV . This is not the case for Sobolevspaces with similar regularity. For nonlinear degenerateconvex fluxes the optimal smoothing effect is proven. Somehints will be given on the non convex case. The importantnotion of nonlinear flux will be discussed. Furthermore,the fractional BV regularity implies the maximal regular-ity conjectured by Lions-Perthame-Tadmor in 1994.
Stephane JuncaUniversite de [email protected]
Christian BourdariasUniversite de [email protected]
Pierre CastelliUniveriste de Nice, [email protected]
Marguerite GisclonUniversite de [email protected]
MS82
Dissipation Vs. Quadratic Nonlinearity: from En-ergy Bound to High-Order Regularizing Effect
We consider a general class of evolutionary PDEs involv-ing dissipation (of possibly fractional order), which com-petes with quadratic nonlinearities on the regularity ofthe overall equation. This includes the prototype mod-els of Burgers’ equation, Navier-Stokes equations, thesurface quasi-geostrophic equations and the Keller-Segelmodel for chemotaxis. Here we establish a Petrowskytype parabolic estimate of such equations which entaila precise time decay of higher-order Sobolev norms forthis class of equations. To this end, we introduce asa main new tool, an “infinite order energy functional’,E(t) :=
∑n ant
n‖(−Δ)nθ/2u(∗, t)‖L2 , which captures theregularizing effect of all higher order derivatives of u(∗, t),by proving — for a carefully, problem-dependent choice ofweights {an}, that E(t) is non-increasing in time.
Eitan TadmorUniversity of [email protected]
Animikh BiswasUniversity of North Carolina
MS83
Nonlinear Diffusion Equation Model of BacterialDynamics
Abstract not available at time of publication.
Mark S. AlberUniversity of Notre DameDepartment of Computational and Applied Mathematicsand [email protected]
MS83
Fully Anisotropic Diffusion Equations and Appli-cations to Glioma Growth and Wolf Movement
Abstract not available at time of publication.
Thomas J. HillenUniversity of AlbertaDepartment of Mathematical and Statistical [email protected]
MS83
Scale Invariance in Mathematical Models of Bio-logical Pattern Formation
Abstract not available at time of publication.
Hans G. OthmerUniversity of MinnesotaDepartment of [email protected]
MS83
Local Kinetics of Morphogen Gradients
Abstract not available at time of publication.
Stanislav ShvartsmanDepartment of Chemical EngineeringPrinceton [email protected]
MS84
Regularity and Uniqueness for a Class of Weak So-lutions to the Hydrodynamic Flow of Nematic Liq-uid Crystals
We establish an ε-regularity criterion for any weak solution(u, d) to the nematic liquid crystal flow such that (u,∇d) ∈Lp
tLqx for some p ≥ 2 and q ≥ n satisfying the condition n
q+
2p= 1. As consequences, we prove the interior smoothness
and uniqueness of any such a solution when p > 2 andq > n.
Tao Huang, Changyou WangUniversity of [email protected], [email protected]
MS84
Existence and Decay of Solutions of the 2dqg Equa-
PD13 Abstracts 123
tion in the Presence of An Obstacle
We study the 2D dissipative quasi-geostrophic equation inthe presence of an obstacle, where the dissipative term isgiven by a fractional power of the Laplacian. Our maintools are the generalized Fourier transform due to Ikebeand Ramm, and the localized version of the fractionalLaplacian due to Caffarelli and Silvestre as improved byStinga and Torrea. We give applications to the problem ofexistence of weak solutions and the decay of these solutionsin the L2-norm.
Leonardo F. KosloffUniversity of Southern [email protected]
Tomas SchonbekFlorida Atlantic UniversityDepartment of [email protected]
MS84
Dynamical Aspects of the Cubic Instability ina Landau-De Gennes Energy for Nematic LiquidCrystals
We consider a four-elastic-constant Landau-de Gennes en-ergy characterizing nematic liquid crystal configurations.It is known that certain physical considerations requirethe presence of a cubic term, which nevertheless makesthe energy unbounded from below. We study the dynam-ical effects produced by the presence of this cubic termby considering the L2 gradient flow generated by this en-ergy. We work mostly in dimension two and our objectiveis to provide an understanding of the relations between thephysicality of the initial data and the global well-posednessof the system.
Xiang Xu, Gautam IyerCarnegie Mellon [email protected], [email protected]
Arghir ZarnescuUniversity of Sussex,[email protected]
MS84
Regularizing Effect of the Forward Energy Cascadein the Inviscid Dyadic Model
We study the inviscid dyadic model of the Euler equationsand prove some regularizing properties of the nonlinearterm that occur due to forward energy cascade. This leadsto several regularity results the inviscid case. We conjec-ture that every blow up has Onsager’s scaling and discussrecent progress toward achieving this goal.
Karen ZayaUniversity of Illinois at [email protected]
Alexey CheskidovUniversity of Illinois Chicago
MS85
Well-Posedness for Variational Wave Equations
We discuss the singularity formation, global existence anduniqueness for energy conservative solutions for variationalwave systems from modelling nematic liquid crystals andother models. Gradient blowup happens in general for so-lutions in these systems, which causes the major difficulty.Especially, a recent progress got with Alberto Bressan onthe uniqueness of energy conservative solution for a varia-tional wave equation
utt − c(u)(c(u)ux
)x= 0, with 0 < C1 < c(u) < C2,
by finding a Lipschitz continuous Finsler metric throughthe optimal mass transformation method is a groundbreak-ing work for nonlinear hyperbolic equations. This presen-tation also includes collaboration works with Ping Zhangand Yuxi Zheng.
Geng ChenMathematics Department, Penn State UniversityUniversity Partk, State College, [email protected]
MS85
Boussinesq System with Nonlinear Heat Diffusionon a Bounded Domain
On a bounded 2D domain with smooth boundary, we provethe global existence of a unique smooth solution to the in-viscid heat conductive Boussinesq Equations with nonlin-ear diffusion, along with homogeneous Dirichlet boundarycondition for temperature and slip boundary condition forvelocity. Such a solution is also shown to be the vanish-ing viscosity limit for the corresponding viscous and heatconductive Boussinesq equation with the same boundarycondition for temperature, and Navier boundary conditionfor the velocity. This talk is based on the joint work withHuapeng Li, Hongjun Yuan, and Weizhe Zhang.
Ronghua PanGeorgia [email protected]
MS85
Title Not Available at Time of Publication
Abstract not available at time of publication.
Mikhail PerepelitsaUniv. of Houston, [email protected]
MS85
Conservation Laws with no Classical Riemann So-lutions: Existence of Singular Shocks
The basic tool in the construction of solutions to theCauchy problem for conservation laws with smooth ini-tial data is the Riemann problem. We review the resultsobtained for the solutions to the Riemann problem andpresent a system of two equations derived from isentropicgas dynamics with no classical solution. We then use theblowing-up approach to geometric singular perturbation
124 PD13 Abstracts
problems to show that the system exhibits unbounded so-lutions (singular shocks) with Dafermos profiles.
Charis TsikkouWest Virginia [email protected]
Barbara Lee KeyfitzThe Ohio State UniversityDepartment of [email protected]
MS86
Competitive Geometric Evolution of ComplexStructures in Fuel Cell Membranes
Surfactant molecules are amphiphilic: composed of a hy-drophilic and a hydrophobic moiety. When mixed witha wettable material the amphiphilic component formsmolecular-width bilayer interfaces and pore structureswhich interpenetrate the bulk regions of the second phase.These structures are manifest as pore networks in blends offunctionalized polymer and solvent, and lipid-filled cellularmembranes in biological settings. The competition for thesurfactant phase generates interfacial dynamics fundamen-tally different from those obtained in mixtures of mutu-ally hydrophobic phases. Using the Functionalized Cahn-Hilliard free energy as a model for amphiphilic mixtures,we employ a multiscale analysis to derive the curvaturedriven flow of smooth closed bilayers and closed-loop porestructures, which we extend to a sharp-interface reductionfor the competitive evolution of collections of bilayer inter-faces and closed-loop pores. In particular, for a mixtureof spherical bilayers and circular closed pores we explicitlyidentify two regimes: one in which spherical bilayers ex-tinguish and the circular pores arrive at a common radius,and a complimentary regime in which spherical bilayers ofdiffering radii stably coexist with common-radius, closed-loop, circular pores.
Shibin DaiMichigan State [email protected]
MS86
Master Equation Model for Charge Transport inDisordered Semiconductors
Charge transport plays a key role in the performance ofbulk heterojunction solar cells. This type of solar cellscontains a mixture of donor and acceptor molecular specieswith a certain degree of phase segregation. I will presenta Pauli master equation model for the charge transport inthese systems and address how the charge mobility is af-fected by the energy offset between the transporting levelsof the two molecular species, the partial volume occupiedby each species and their typical domain size. Our resultsalso pertains partially crystalline samples made of a singlematerial.
Jose FreireUniversidade Federal do Parana, [email protected]
MS86
Hall-Mhd and Related Plasma Problem
In this talk, I will present some derivations and analysis of
the the Hall Magneto-Hydrodynamic equations. We firstprovide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set ofscaling limits. We also propose a kinetic formulation forthe Hall-MHD equations which contains as fluid closuredifferent variants of the Hall-MHD model. Then, we willpresent some well-poseness results for the incompressibleresistive Hall-MHD model, including global weak solutions,local existence of smooth solutions for large data, globalsmooth solutions for small datas well, as a Liouville the-orem for the stationary solutions. We use the particularstructure of the Hall term which has zero contribution tothe energy identity. Finally, we discuss particular solutionsin the form of axisymmetric purely swirling magnetic fieldsand propose some regularization of the Hall equation.
Jian-Guo LiuDuke [email protected]
MS87
Eventual Self-Similarity of Solutions for Diffusion-Absorption Equation with a Singular Source
We consider a class of diffusion-absorption equations witha singular source in Rn. We establish existence of a weakself-similar solutions for these equations which arise in alimit of an infinitely strong singular source. We proveuniqueness of such solution in a suitable weighted en-ergy spaces. Moreover, we show that the obtained self-similar solutions are the long-time limits of the solutionsof the initial value problem for considered class of diffusion-absorption problems with zero initial data and a singularsource of arbitrary constant strength. This is a joint workwith Cyrill Muratov.
Peter GordonDepartment of MathematicsThe University of [email protected]
Cyrill B. MuratovNew Jersey Institute of TechnologyDepartment of Mathematical [email protected]
MS87
Front Propagation in Sharp and Diffuse InterfaceModels of Stratified Media
We study front propagation problems for forced mean cur-vature flows and their phase field variants that take placein stratified media, i.e., heterogeneous media whose char-acteristics do not vary in one direction. We consider phasechange fronts in infinite cylinders whose axis coincides withthe symmetry axis of the medium. Using the recently de-veloped variational approaches, we provide a convergenceresult relating asymptotic in time front propagation in thediffuse interface case to that in the sharp interface casefor suitably balanced nonlinearities of Allen-Cahn type.The result is established by using Γ-convergence type argu-ments to obtain a correspondence between the minimizersof an exponentially weighted Ginzburg-Landau-type func-tional and the minimizers of an exponentially weightedarea-type functional. These minimizers yield the fastestmoving traveling waves in the respective models and de-termine the asymptotic propagation speeds for front-likeinitial data. We further show that generically these frontsare the exponentially stable global attractors for this kind
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of initial data and give sufficient conditions under whichcomplete phase change occurs via the formation of the con-sidered fronts.
Annalisa CesaroniDipartimento di MatematicaUniversita di [email protected]
Cyrill B. MuratovNew Jersey Institute of TechnologyDepartment of Mathematical [email protected]
Matteo NovagaUniversity of Padova (Italy)[email protected]
MS87
Quasi-Static Evolution and Congested Crowd Mo-tion
In this talk we investigate a transport equation with a driftpotential, where a constraint on the L∞ norm is imposedon the density. This model, in a simplified setting, de-scribes the congested crowd motion with a density con-straint. When the drift potential is convex, the crowddensity is likely to aggregate, and thus if the initial den-sity starts as a patch (i.e. if it is a characteristic functionof some set), then the density evolves like a patch. Weshow that patch evolves according to a quasi-static evolu-tion equation, which is a free boundary problem and hassome connection with the Hele-Shaw equation. To showthis result we make use of both viscosity solutions theoryas well as the gradient flow structure of the problem.
Damon [email protected]
Inwon KimDepartment of [email protected]
Yao YaoDepartment of MathematicsUniversity of [email protected]
MS87
Discrete Motion by Mean Curvature
We will discuss some results on the motion of interfacesin a discrete environment. The model takes into accountnearest and next nearest neighbor spin interactions lead-ing to interfaces between patterns. Intricate stick-slip phe-nomena also arises through different spatial and temporaldiscretization. This is joint work with Andrea Braides andMarco Cicalese.
Aaron YipPurdue UniversityDepartment of [email protected]
MS88
Well-Posedness and Asymptotic Stability for the
Lame System with Infinite Memories in BoundedDomain
In this work, we consider the Lame system in 3-dimensionbounded domain with infinite memories. We prove, un-der some appropriate assumptions, that this system is wellposed and still stable, and we get a general and preciseestimate on the convergence of solutions to zero at infinityin term of the growth of the infinite memories.
Ahmed BchatniaUniversity of [email protected]
MS88
Vanishing Viscosity Limit of Some SymmetricFlows
Abstract not available at time of publication.
Gung-Min GieIndiana [email protected]
MS88
Singular Perturbation Analysis for Convection-diffusion Equations with Corners
We study the asymptotic behavior at small diffusivity ofthe solutions to a convection-diffusion equation in a rect-angular domain ?. The diffusive equation is supplementedwith a Dirichlet boundary condition, which is smooth alongthe edges and continuous at the corners. To resolve the dis-crepancy between the exact solutions and the correspond-ing limit solution on the boundaries, we propose asymp-totic expansions of at any arbitrary, but fixed, order. Inorder to manage some singular effects near the four cornersof ?, the so-called elliptic and ordinary corner correctors areadded in the asymptotic expansions as well as the parabolicand classical boundary layer functions. Then, performingthe energy estimates on the difference of the exact solutionsand the proposed expansions, the validity of our asymp-totic expansions is established in suitable Sobolev spaces.
Chang-Yeol JungUlsan National Institute of Science and TechnologySouth [email protected]
MS88
How Do Flows in Conduit and Porous Media In-teract?
Multiphase flow phenomena are ubiquitous. Common ex-amples include coupled atmosphere and ocean system (airand water), oil reservoir (water, oil and gas), cloud and fog(water vapor, water and air). Multiphase flows also playan important role in many engineering and environmentalscience applications. In some applications such as flows inunconfined karst aquifers, karst oil reservoir, proton mem-brane exchange fuel cell, multiphase flows in conduits andin porous media must be considered together. How freeflows in conduit/channel interact with flow in porous me-dia is a challenge. In this talk we present a family of phasefield models that couples two phase flow in conduit withtwo phase flow in porous media. These models together
126 PD13 Abstracts
with the associated interface boundary conditions are de-rived utilizing Onsager’s extremum principle. The modelsderived enjoy physically important energy law. Numericalscheme that preserves the energy law will be presented aswell.
Xiaoming [email protected]
