This report was prepared as am account of work sponsorrd by an agency ot the Uniteti Statu Government. Neithet the United States Government nor any agency thercOt, nor m y of the2 cmployets, n??keS any m t y , express or implied, or ass om^ aky legal IiabGty or nsponsiblkt~ for the accuracy, c o m p l t c n ~ or use- fubess of any information, apparatw, product, -or ptoctss d&oscd,-or rtprrtcnts that its use would aot infringe priratdy owned nata Reference b e r m to any spe- ciftc commercial product, proccst, or d c e by trade name, tradematlt, manufac- turer, or otherwise docs not necessarily constitute or imply its endonemen~ m m - meadzdion, or favoring by the United State Government ot any agency t h d . The views and opinions of authors txprrsscd hacia do not n d y state or reflect those of the United Statu Government or any agency thmf .

,

Portions of this document may be il!egible. in electronic image products. Images are produced from the best available original document.

Nonlinear Analysis 2000 - This oonfennce is dedicated to the state of the art in nonlinear analysis and its applidons,

with emphasis on partial and onlinaq digmntial equations. ApplicotioM will include n u d cnl analysis, OptimaI wntml, inverse pmbkms, mathemaw physics, dynomicol systemsl fluid dynamics, mothemntical biology’ mathenuatid finance, and other anxu of applied mathematics.

Our god is to bring together the rising gmemtion of seicntisb fnnn all over the world; to let them highlight a sample of their wrfc, meet ench other, and talk science.

The wnfennce is dedbted to to sc imtbb who an early in their c u m . Then will be about 60 tolkc and 1W posters prwmted during the meek; all am by gmduate students, postdoes, or junior faculty. In addition thcn will be open discussion sur ioM and small workisrg gmups to discuss to- and apprrnchu in mme detail.

Due to spaoe limitations, participation in the mfermce had to be limited to about 150 seien- tists. Nearly 600 had applied.

Organizing Committee Yan Guo Brown University Bob Jerrard University of Illinois at Urbana-Champaign . Thomas Kriecherbauer Universitat Munchen, Germany Norbert J. Mauser Universitat Wien, Austria Mary Pugh University of Pennsylvania

Scientific Committee Organizing Committee and Carson Chow University of Pittsburgh J&a Cvitanic George Haller Brown University Enrique Pujals IMPA, Brazil Leonid Ryzhik University of Chicago Housnaa Zidani

David McLaughlin Courant Institute, NYU

Columbia and University of Southern California

Universite d’Orleans and INRIA, France Direct or

Conference Secretary Carol Levine

Funding: Courant Institute New York University Institute for Mathematics and its Applications Department of EnBrgy Office of Naval Research Army Research Office Air Force OfRce of Scientific Research

Cover design and photograph by Jun Zhang, Courant Institute - The photogmph show the wake of a thin soap flm powing past a pezible filament. The -+ ment w perfonned by Jun in Coumnt’s Applied Mathematics Lobomtory.

Locations

M-xxx: Main building: 33 Washington Place M-703: Main building, 7th floor, lecture room 703 M-713: Main building, 7th floor, lecture room 713 M-714: Main building, 7th floor, lecture room 714 M-HH: Main building, 1st floor, Hemmerdinger Hall C-xxx: Courant Institute, 251 Mercer St C-L13: Courant Institute, 13th floor, lounge C-101: Courant Institute, 1st floor, lecture room 101 (2-102: Courant Institute, 1st floor, lecture room 102

Cantor: Cantor Theatre, 36 East 8th St

Sunday May 28, 2000

1030-11:40 I registration on 7th floor of Mein 11:40-1200 I opening remarks (in M-703) I by Dave Mchughlin and organizers

talks (in M-703)

1200-12:15 S h a n k Venkataramani, "What does one learn from a crumpled sheet?" 12:20-12:35 hancois Oust ry , "On the HartresFodr variational problem: NP-hardneas and

12:40-1255 Petcr Miller, "Semiclassical Asymptotiar for the Focusing Nonlinear Schrijdinger

1:00-1:15 Jinho Baik, "Symmetrized random permutations and random matrix theory"

semidefinite relaxation"

Equation"

1:20-210 pzzzq

talks (in M-703)

2:10-225 W r i c Villani, "On the ation for long-range interactions" 230-245 Aaron N. K. Yip, "Noise and Uniquenets of Motion by Mean Cumture" 2:50-5:05 M a r h Katsoulakis, "A multiscale approach to cluster growth problems" 3:10-a25 Carlo Mantegazza, 'smooth Geometric Evolutions of Hypersurfaces"

ITEiJ stretch legs, put up posters in M-HH 33w:oo 4:oo-s:oo topic poster-presentation/discussion groups:

"fluids" (in M-713), moderated by Wendy Zhang and Tom Witelski "dynamical systems" (in M-714), moderated by Vadim Kaloshin and Bj6m

Sandstede.

s:o0-6:00

6:15-7:45

poster session (in M-HH) (soda provided)

conference reception l i n the Courant lounge (CL13)

Weizhu Bao

Sahraoui Chaieb

David Edwards

Petri Fast

Angela Jimenez-Catm

Boualem Khouider

Igor Kliakhandler

Lou Kondic Peter Mucha

Adam O b e r m a n

Chris Wiggins

Tom Witelski Wendy Zhang

PierreAntoine Absil

Jean-Baptiste Caillau

GBbor C s e d k

vassill Gelfreich

Anna Gilber t

JBrg Hlirterich

Ning Ju

Vadim Kaloshin

Thomas Kriecherbauer

James Robinson

Ralf Wit tenberg

Posters in Sunday's session:

T h e random projection method for hyperbolic conservation laws

"Mixing Immiscible Fluids: Drainage induced cusp formation"

"Surface Reaction Near a Stagnation Point"

"An overset grid method for viscous fingering in non-Newtonian

"Asymptotic Behavior of a coupled ODE/PDE system for a c l o d

"Numerical combustion via an asymptotic flamelet library"

"Inverse cascade in film flows"

"Instabilities in the flow of thin liquid films" "Fluctuations and Structures in Dilute Sedimentation"

"Geometric Equations for Reaction-Diffusion equations"

"Dynamics of a 1D Flag in a 2D Wind" "Finite-time seK-similar rupture of thin fiIms" "Rod climbing in nematic liquid crystals"

"A generalized Rayleigh quotient iteration for computing invariant

"Cantinuation technique for a weakly controlled satellite"

"Methods for the Estimation of the Life Expectancy of T h s i e n t

"Asymptotiar beyond all orders near Hamiltonian bifurcations" "A random dynamical system model for TCP" "Singular Perturbations and 'hveling Waves of Conservation Laws

with Sourcea"

"Dishinguished Hyperbolic 'hjectories of Aperiodic Flows and Approximation"

" A HawdorfT dimension is not preserved under Embeddings of At- tractors from Infinite-Dimensional into Finite-Dimensional Spaces and B Growth of number of periodic points for generic diffeomorphisms"

with stiff reaction terms"

fluids"

thermosyphon"

Subspaces"

Chaotic Behaviour"

"On the dynamics of nonlinear lattices"

"Determining flows using a finite number of point obse+ions" "Spatiotemporal Chaos and the Kuramot4ivaahinsky Equation"

".I x.I ~~ . . ... ..... .. l_l" " -

Monday May 29, 2000

talks (in M-703)

900-9:15 Bruno Bouchard, "Option Pricing via Utility Maximization in the presence of

920-9:35 Ronnie Sircar, "Modeling of Market Volatility" 9:40-9:55 Rama Cont "Random fields, stochastic partial differential equations and interest

. 10:00-1015 Paul Batcho, "High Precision Calculation of Electronic Structure for Quantum

'hnsaction Costs: an Asymptotic Analysis"

rate model"

Chemistry"

10:20-1050 1-1 10:20-1050 1-1

talks (in M-703)

1050-11:20 Jun Zhang, "Dynamic boundaries in fluid flows, an experimental study" 11:30-11:45 Ricardo Cortez, "Computation of b a t Immersed Boundary Motions" 11:50-12:05 Heiko von der Mosel, "Modeling DNA-molecules"

talks (in M-703)

1:50-205 210-225

2:30-245

2110-3:05

Bjijrn Sandstede, %ability and instability mechanisms of spiral waves" Werner Kistler, "'hveling waves of excitation in a 2-D network of spiking neu-

Sima Setayeshgar, "Electrical Wave Propagation in the Heart: The Dynamics of

Rancesca Da Lio, "A Strong Comparison Rosult for Quasilinear Equations in

rons"

Scroll Waves in Anisotropic Excitable Media"

Annular Bounded Domains and Applications"

5:10-3:30 [break1 stretch legs, put up posters in M-HH

3:30-4:30 topic poster-presentation/discussion groups: "mathematical biology and neuroscience" (in M-703), moderated by Sonya

"nonlinear optics and integrable models"(in M-713), moderated by Jared

"mathematical finance" (in M-714), moderated by Ronnie Sircar.

poster session (in M-HH) (soda provided)

Baha r and Carlo Laing,

Bronski and Pcter Miller,

4:3&5:45

5:45-E30 -1 E30-9:00 discussion with soda, beer, and wine in C L 1 3

Sonya Bahar Victoria Booth Amitabha Bose

Duncan Callaway Alain Goriely

Kresimir JosiE Carlo Laing Douglas Mar Georgiy Medvedev

Anita %do-Goriely Simon Schulte Alexei Borodin Annalisa Calini

Ibrahim Fatkullin Josselin Garnier

Posters in Monday's session:

"Model for a Ribbon Synapse in the Paddlefish Electroreceptor" "Hippocampal Place Cells and the Generation of Temporal Codes" "Using synaptic depression to switch between distinct oscillatory

"A Simple Model of Epidemics with Pathogen Mutation" "Homoclinic collapse and bistability in the statics and dynamics of

"Chaotic Phase Locking in Theory and Practice" "Bump Attractors in Networks of Spiking Neurons" "Noiseshaping and anti-synchrony in model neuronal networks" "Synchronization and transient dynamica in the chains of electrically

coupled FitzHugh-Nagumo oscillators" "Mathematical Models of Ionic Diffusion in Olfadory Glomeruli" "Entropy and information in neural networks" "Riemann-Hilbert problem and the discrete Bessel kernel" "Connecting geometry, topology and spectra for finitegap N U

modes"

biological filaments"

potentials" "Goursat problem for Maxwell-Bloch equations" "Inverse F t t e r i n g transform technique for solitons in random

"Coupled Mode Equations in Nonlinear Fiber Optics" "A mean-field statistical theory for the nonlinear Schriidinger

"Effects of Discreteness on Stability Problems" "Adventures of Solitary Waves in D i m t e Worlds" "Relativistic Dust Disks and Hyperelliptic Riemann Surfaces" "Logarithmic potential theory in singular limits of integrable

media"

equation"

SyStems"

Roy Goodman Rich Jordan

Todd Kapi tu la Panayotis Kevrekidis Christian Klein Ken McLaughlln

NicolaeCoriolan Panoiu

Keith Promislow

Constance Schober

Stephanie Singer David Trubatch

Bjorn Walther . Jerome Busea Craig Friedman

Henri Schurz

"Adiabatic perturbative analysis of two-soliton interaction with a p

"Modulational Stability via Renormalization Group for the Optical

"Long time dynamics of the modulational instability of deep water

"Group Theory ( a k a . Symmetry) and Differential Equations" Yntegrable Discretization of the Vector Nonlinear Schriidinger

TBA "Convergence Issues in Evolution Equations" "Contingent Claim Pricing and Hedging under Multiple Probability

"General principles for numerical approximation of nonlinear s b

plications to soliton propagation in optical fibers"

Parametric Oscillator"

waves"

Equation"

Measures on a Finite Set of Monte Carlo Paths"

chastic differential equations"

Tuesday May 30, 2000

talks (in M-703) 900-015 Alex Kiselev, "Bulk Burning Rata in Pasive-Reactive Diffusion" k20-935 Eric Smi th , "Self-starting heat engines as dynamical critical systems" 9:40-9:55 Leonid Koralov, "Moment and Almost Sure Lyapunov Exponents for the Solution

of the Parabolic Anderson Problem" 1000-1015 Tomasz Komorowski, "Diffusion Approximation For Solutions of Convection-

Diffusion Equation with a Random Drift"

1020-1O:SO

talks (in M-703)

1050-1120 Itai Cohen, "Exploring Bansitions in Two Fluid Interface Systems" 11:SO-11:45 M a m Fontelos, "Some Results on the Evolution of Thin Non-Newtonian Jets" 11:SO-1205 Ovidiu Coatin, "New analytic methods for PDEs in the complex domain"

1210- * 1 free afternoon and evening 1 There will be some informal events arranged.

Wednesday May 31, 2000

. talks (in M-703)

9:OO-9:15 Chongchung Zeng, "Homoclinic orbits for near-integrable PDEe" 9:20-0:35 Marcel0 Viana, "Chaotic dynamics from a probabilistic viewpoint" 9:40-965 Sanjeeva Balasuriya, "Eddy growth criteria via a M e l n i b approach" 1O:OO-1015 Boyan Sirakov "Harnack type estimates for cooperative elliptic systems"

1020-10:50 1-1

talks (in M-703)

1O:SO-11:20 Randall Kamien, "Minimal Surfaces and Topological Defects" 11:SO-11:45 Silvia Serfaty, Vorticity for the Ginzburg-Landau equation of superconductivity" 11:50-1205 Barbara Nie thammer , "Macroscopic models for Ostwald Ripening"

1210-1:50 1-1

talks (in M-703) 1:50-2:05 210-225 230-2:45

2 5 0 4 0 5

Knut Selna, "Acoustic pulse shaping and localization in a random fractal" Sibel 'hri, "Nonlinear Diffusion-in Computer Vision" Liliana Borcea, "Nonlinear Multigrid for Imaging Electrical Conductivity and

Andreas Miiller, "A Parallel Distributed Climate Model for Precipitation" Permittivity ,at Low Frequency"

3:10-330 lbreakl stretch legs, put up posters in M-HH

3:30-4:30 topic poster-presentation/discussion groups: "fluids"(in M-713), moderated by Pete K r a m e r and Marcel Oliver ucalculus of variations and mater ia ls science" (in M-714), moderated by

Umathematical physics and modelling" (in M-709), moderated by David

poster session (in M-HH) (soda provided)

Barbara Nie thammer and Felix O t t o

C a i and Eric vanden Eijnden.

4:30-5:45

730-9:00 discussion with soda, beer, and wine in GL13 I

Martin Bazant '

RaphGl Danchin

Zoran G r a i b

Peter K r a m e r

Rick Laugesen

Marwan Moubachir

Manx1 Oliver

Michael R f Z i E h

Xiaoming Wang

Craig Zirbel Georg Dolzmann Carlos Garcfa Cervera Bernd Kirchheim

Christopher Larsen Dawn Lott&rumpler

Regis Monneau Luisa Moschini

Matteo Novaga Petr Pl&E

Daniel Wienholtz

Xiaodong Yan

Meredith Be t t e r ton

Radica Costin

Marco Cza rnsd t i

Klaus Hornberger

Cyrill Mura tov

John Pelwko Pierpaolo Soravla

Lourdes Tell0

Posters in Wednesday's session:

"Asymptotic Analysis of Electrochemical Diffuse Charge Layers"

"Zero Mach Number limit in critical spaces for compressible Navier-

'The geometric structure of the vorticity super-level sets and regu-

"A Simulation Method for Thermally Fluctuating Fluid Systems

"'Thin Fluid Film' PDEe - Stability and Energy Levels of Steady "Instability 'Ikacking via Optimal Control" "The vortex blob method aa a second-grade non-Newtonian fluid" "Electrorheological Fluids Modeling and Mathematical Theory"

"Boundary Layer Associated with Incompreasible Flows"

"Random velocity Relds with known Lagrangian law" "Microstructure and quasiconvex hulls of seta" "Micromagnetics of Thin Films" "Gradients without r d - o n e connections" "Optimal design with perimeter penalization" "A numerical technique for optimal patterns for suturing wounds of

"Some Regularity Ftesulta for the Obstacle Problem" "On a Class of Semilinear Dirichlet Problems with First Order

"A variational problem in crystalline evolution" "Numerical approx,imation of macroscopic quantities in the presence

of microstructure in elastic solids" "Branch points of minimal surfaces"

"Nonsmooth minimizers of smooth strongly convex function&" "Formation of Penitentes"

"Nonintegrability criteria for a clasa of diffemntial equations with

"Asymptotic control and stabilization of nonlinear d l l a t o r s with

T h e interior and exterior spectrum of magnetic billiards" "Collective dynamics of M n g patterns in reactiondiffusion

YLOcal and Nonlocal Problems in Modeling of MEMS Devices" "Boundary value problems for Hamilton-Jacobi equations of optimal

"Infinitely many stationary solution8 for a simple climate model"

Stokes equation"

larity for 3D Navier-Stoh equations"

with Immersed Structures"

States"

complex shapes to foster healing"

Tern"

two regular singular points"

non isolated equilibria"

systems"

control with discontinuous running m t "

Th-sday June 1, 2000 Posters in Thursday's sessiow Deborah Al te rman "Diffractive short pulse asymptotics for nonlinear wave equations"

David Ca i 'Dispersive Wave Turbulence in One Dimension" Alina Chertock "Self-Similar Intermediate Asymptotics for a Degenerate Parabolic

Filtration-Absorption Equation"

Jim Colliander "Bilinear Estimatea and Applications to Nonlinear Schtijdinger Equations on R2"

Gustavo Crurr-Pachea, "Coherent structures on a perturbed Nonlinear Schroedinger Equation"

Stephen G w t a f s a n "Equivariant solutions of Ginzburg-Landau equations: stability and interaction"

Julie Levandosky "Smoothing Properties for Nonlinear Dispersive &UatiOnS"

MartaLewicka "Stability of patterns of large non-interacting waves" Yi Li "Linear Stability OF Solitary Waves of a Nonlinearly dispersive Hamil-

Felipe L i n a m 911-posedness for the Derivative Schtijdinger equation"

Yue Liu "Blow up and instability of solitary-wave solutions to a generalized

Judith Miller "Dispersive effect0 in a modified Kuramot&ivashinsky system" Chongsheng Cao "Asymptotic Behavior of visa~us I-D Scalar Conservation Laws with

Neumann Boundary Conditions"

Hends El Fbkih "Asymptotic analysis for some control problems" Peter Howard "Pointwise estimates and stability for degenerate viscous shock

Marianne Korten "On the Geometry of the Free Boundary of the onephase Stefan

Maria Belen Lerena "A parabolidliptic equation arising in the transient regime of a

John McCuan "Examples of quasiconcavity via the maximum principle"

Hosein Tehran1 "On a Class of Asymptotically Linear Elliptic Problems in P" Elide Terraneo "Non-Uniqueness for a Nonlinear Heat Equation"

tonian System"

Kdomtsev-Petvlashvili equation"

waves"

Problem"

magnetically confined plasma in a Stellarator"

talks (in M-703)

900-9:15 Rubh Spias, "Parameter Identification in Nonlinear Abstract Cauchy Problems * Using Quasilinearization"

020-935 Covind Menon, "Kinetics of materials with wiggly energiea and related averaging

940-958 Hongkai Zhno, "Moving Interface Problems, Numerical Methods and Appliw

10:OO-1015 Michael Brenner , "Attraction and Diffusion with Hydrodynamic Interaction"

problems"

tions"

1020-1050 -1

talks (in M-703)

1050-11:20 Daniel Lee, "Biologically Inspired k i n g Algorithms" 11:90-11:45 Stephen Coombes, "Cellular signalling: Composing global signals from elemen-

11:50-1205 Carl van Vrureswijk, "Stability of the ABynchronous State in Networks of Strongly tary events"

Coupled Oseilhtors"

13:10-1:50 1-1

talks (in M-703)

1:50-2:05

210-225

230-245 250-3:05

Axel Osses, "Approximate controllability and homogenization of a semilinear el-

Thomas Bewley, "DNSbased predictive control of turbulence: an optimal bench-

Jared BronsM, 'Taasive scalar intermittency and the moment problem" Yuri Lvw, Weak turbulence theory: development and novel applications"

liptic problem"

mark for feedback algorithms"

3:10-3:30 /breakj stretch legs, put up pcsten, in GL13

3:30-4:30 topic poster-presentation/discussion groups:

liander and Konstantina lkivisa

and Toti Daskalopodos

"dispersive a n d hyperbolic problems " (in C-101), moderated by J i m Col- "elliptic a n d parabolic problems" (in C-102), moderated by Jose Carrillo

490-545 poster session (in GL13) (soda provided)

7:00-1000 . I conference banquet in Chinatown 1

g:O0-916 9:20-935 9:4&955

1000-1O:lS 1020-1035

Friday June 2, 2000

talks (in Cantor) Robert McCann, "Kinetic Equilibration Rates for Granular Media" Rancois Hamel, "Front propagation phenomena in periodic media" Hatem Zaag, "A global approach for the study of blow-up in nonlinear heat qua-

Jose Carrillo, "Entropy didpation method for parabolic equations" Pierre-Emmanuel Jab in , "Control on the distribution function of particles in a

tions" .

S t o h flow"

11:10-11:25

31330-1h45

11:60-12x05

1210-12x25

12:30-12:45

talks (in Cantor)

Gigliola Staffllani, "Global Wel-posednesa for dispersiw equations with rough

Konstantina Mvisa, "On the L' Well-@nesa of Solutions to Hyperbolic S y e

Maria Psarelli, "Maxwell-Klein-Gordon Equations in 4-dimensional Minkowski

Alexander K u r g a n w , "New High-Resolution Central sdremea for Conservation

Mdria Luk6Eod-Medvfdov6, "Evolution Galerkin schemes for multidimensional . hyperbolicsystem"

data"

t e m of Conservation Lam with Large BV Data"

Space"

Laws, Hamilton-Jacobi Equations and Related Problems"

12:50-1:00 I closing remarks by Dave McLaughli ]

ITalks on Sunday May 28, 20001

What does one learn from a crumpled sheet? Shankar C. Venkataramanl P

University of Chicago, Department of Mathematics shanka&math.uchicago.edu

Everyday experience shows that a sheet of paper will awnple when it b crushed. Crumpling h an ubiquitous phenomenon that OOEUM in a variety of systems, ranging fium the membranes of vesicles in living cella to the shells that are wed in engiaearing and packaging. A basic puzzle b the following: Why doos a sheet of paper crumple when it is crushed, that is, why does the geometry of the sheet become rough on the scales of the applied stress?

I will talk a b u t an analysis of thb question and it's genedimtion where in one considers the crumpling of a m-dimensional sheet in a d d i i i o n a l spam, for general m and d I will present some of the results on this probkm from joint work d t h Brian DiDonna, Tom Wtten, Eric KrMler and Bob Gemch. I will dso outline some of the open issues in the analysb of this problem. Finally, I will talk about how this problem relates to many interasting problem in diverse fields including Differential geometry, Optimization and the c.lculus of variations, Materid science and Gauge Field theories.

ardness and semideflnite relaxation

Joint work with

In this talk we show that computing a solution for the Hlutree-Fock variational problem (q) of an N-fermions system is NP-hard. The proof b obtained in two : (1) we start from a quadratic formulation, due to C. Clurod and J. K. Pereus, in the space of one-body and *body density matrices ; (2) then we ptwe that there exists a polynomial transformation from the diagonal (AF) to the MAXCUT problem. Using Lagrangian dudity we dso derive a convex semideflnlte relaxation for 0.

Semiclaasical Aaymptotlca for the Focusing Nonlinear Schriidinger Equation Peter D. Miller

Department of Mathematiclr and Statistics, Monssh University millerpd~msil .mathmon~h~u.au

Some aspeas of joint work with S. Kam- and K. T.-R. MeLsughlin on the asymptotic analysis of a Riemann-Hilbert problem corresponding to the dc lass ica l limit of the in-scattering solution of the focusing nonlinear %&linger equation will be presented. Our main result b a rigomus proof that the s e m i d d d l i t for particulsr initial data considered by Satauma and Yajima in 1974 b indeed governed for a short time by the elliptic Whltham modulation equations. COmputatbn of the limit, including obtaining euct solutions of the Whitham equations correspond^ to general initial data, can be interpreted in term of the critical point theory of an electrostatic w r g y functional.

Symmetrized random permutations and random matrix theory Jfnho Balk

Princeton University and Us j baiWmath.princeton.edu

Censider a Poisson prowas in the plane. We are interested in the length of the longest up/right path from (0,O) to (t,t) M t gets large. It wlil be p-ted that the limiting fluctuation is identical to that of the largest eigenvalue of a random GUE matrix. We alm impose varbus symmetry conditions on the Poisson pmcess, and show that the limiting fluctuation Is now equal to that of the largest eigenvalue of a random GOE/GSE matrix. This problem arlse~~ in several placed including random pennutationq random viclow wnlb and polynuclear growth models.

On the Baltzmann equation for long-range Interactions CMriC VUIanl

&ole Mrmale supMeure, DMA, Paris villaniQdmaens.fr

Thls talk will be devoted to the Boltzmann equation when particles intupct through long-range inter- actions. In this ean, Grad's angular cutoff aglumption h not realistic, due to the abundance of grazing collisions, and for a very long tims this h~ beem a severe limitation to the mathematical understanding of this equation. Here I Id present recent results on the subject, obtained in collaboration with Alexan- dre, among which b the 6mt theorem of exismco of weak mlutions, and the pmof of a smoothing effect conjectured by Lions several yean ago. The pmob rely c ruddy on a 6ne study of the entropy dissipation smoothing, whieh I performed with Alexandre, Desvillettee and Wennberg. Not only do grazing Cdi i s iOnS entail huge mathematid technicalities, but they have a considerable influence on the qualitative behavior of solutions.

Noise and Uniqueness of Motion by Mean Curvature A w n N.K. Yip

Purdue University, Department of Mathematics yipQmath.purdue.edu

We will discw the issue of non-uniquenear in the motion by mean curvature of a surf-. Such a geometric motion b a simplied model for solidillcation proasmu We will demonstrate how noise can be used to eliminate the non-uniquenegl phenomena and thus provide a selection principle for the surface evolution.

A multiscale approach to duster growth problems M a r b Katsoulakb

Unlvemity of Massachusety Amherst markosQmath.umaas.edu

We present a syst8tnrtic appro& to derive menoccopic theories (Le. stochastic integrodiRemntial equations) and macroscopic pverning lam of gr&vth vebdty and morphological evolution of dusters, obtained directly h m microscopic stochastic systems. Exampler presented include surf- and deposition P-.

We 6mt introduce the micmcopic mechanbm at a &atistical mehmics Level, which descrlbe the patticle-by-partide formation and ewlution of clustem; theM modeb -tidy constitute Monta cario algorithms for the phenomena under consideration, and since they am computationally intensive, are

.

http://shanka&math.uchicago.eduhttp://baiWmath.princeton.eduhttp://villaniQdmaens.frhttp://yipQmath.purdue.eduhttp://markosQmath.umaas.edu

suitable only scribinn short d e s . Often. rs is the c ~ a e deposition models, the microscopic proceww need to be coup& to a continuum m&ia model at a much larger space/time Scale. mi underscores the need to bridge the discrepancy between the miero- and macro- models by deriving m e scopic and macroscopic PDE for the evolving clusters, directly from the microscopic particle s~ystems. One of the crucial steps here b the identiaeatin of macroscopic quantities (e.g. tmnsport mfichts, surface tension, etc.) in terms of the microscopic parameters, i.e. interaction potentials and type of dynamica

We finally discuss the microscopic validation of the memxopic and macroscopic picture through gradi- ent Monte Carlo simulations, and present spectral numerical methods for the derived mewcopic models.

Smooth Geometric Evolutions of Hyperaurfacea Carlo Mantegsrsa

Scuol. Normale Supetiore di Pisa rnantega@sns.it

We consider the gradient flov d a t e d to the following functionals

Fm(P) - lM 1 + IvVZ c . The function& are defined on hypersurfaces immersed in Rn+' via a map (p : M + R"+', where M is a smooth closed and c o d n-dimensional manifold without boundary. Here p and 0 are respectively the canonical measure and the kvi-civita connection on the Riemannian manifold (M,g), where the metric g is obtained by pulling b d on M the usual metric of R"+' with the map 9. The symbol V"' denotes the m-th iterated covariant derivative and Y b a unit normal 1 0 4 vector field to the hypermuface.

Our main d t is that if the order of derivation m E N is strictly larger than the intew part of n/2 then singularities in finite time cannot occur during the evolution.

These geometric functiolula are related to similar ones p r o p o d by Ennio De Ciorgi, who conjectured for them an analogow regularity result.

ITalks on Monday May 29, 20001 I - v I

Option Pricing via Utility Maximization in the presence of 'Ikanaaction Costs: an Asymptotic Analysis

Bruno Bouchard CEREMADE, Universite Paris Dauphine, and CREST

bouchard&ensae.lr

We consider a multivariate financial market with proportional transaction costs as in Kabanov (1999). We study the problem of contingent claim pricing via utility maximization ea Q Hodgea and Neuberger (1989). Using an exponential utility function, UB derive a cloned form ehsracterization for the rsymptotic price as the risk aversion tends to infinity. We prove that it b reduced to the super-replication cost if the initial endowment is only invested in the non-risky asset, as it was coqjectured In Barles and Soner (1996). In the bmwnian diffusion CO(LB, we do not make use of the dual formulation for the super-replication price obtained in Kabanov (1999). Finally, we explain how this results may be generelized to semi-martingales.

Modeling of Market Volatility Ronnie Slrur

Department of Mathematics, University of Michigan sirdurnich.edu

In modern financial markets, investors and trading institutions are faced with an environment of uncertain and changing volatility which must be modeled when pricing or managing the risks from holding derivative securities. However the situation is complicatcd becnuse volatility is not directly observable

We describe an asymptotic and statistical analysis of the problem that exploits the tendency of volatil- ity to come In bursts, or cluster. A singular perturbation approach provides a correction to the Biack- Scholes pricing and hedging theory that adjusts for random volatility in a robust way, in that it does not depend on specific modeling of the volatility prcoess. We illustrate how the modeling assumptions fit SkP 500 index data, and how the corrected theory identifies stable market quantities that are easily estimated from the o b s e d implied volatility surface.

Joint work with Jean-Pierre Fbuque, George Papanicolaou and Knut Solna

Random fields, istocherrtic partial differential equations and interest rate modeling Runs Cont

RamaContBpolytechnique. fr Centre de Mathht iques Appliqudes Ecole Polytechnique, Ran=.

Interest rates are naturally described in terms of a random field t(t,z), inde xed by tim e and the maturity of the rates. Commonly used interest rate models belong to the Heath-Jarrow-Morton class which can be described in tem of a hyperbolic stochastic PDE for the random field r(t, z). The empirical analysis of interest rata fluctuations [I, 2,6] mveaki interesting and non-trivial statistical foatures of thii random Md, which HJM models fail to capture in a simple manner 141.

Viewing HJM models ea inki te dimensional diffusions in a suitable Hilbert spac e, we address some of the inadequacies of HJM models and propose alternative approach which models the forward rata term structure OS an infinite dimensional Ornstein Uhlenbeck process. Via Sturm-Liouviile theory and using the hypothe& of "market segmentation", the forward rates t ( t ,z) can then also be interpreted as the solution of a pambolic stochastic PDE, of which we study some simple CBB~S in detail.

We discuss the properti- of the solution8 and show that they capture in a parsimonious manncr the essential features of yield curve dynamica: imperfect correlation between maturities, mean rewrsion of interest rates and the structure of principal components of term structure deformations. We then study extensio~ of this basic model to more complicated volatility structures and discuss the duality between the choice of the volatility structur e and the choice of the state space [5].

Keywords: stochastic PDE, Sturm-LiouvUe problem, cylindrical brownian motion, infinite dimensional stochastic differential equations, random fields, tarm structure of interest rates, forward rates, random strings, multifactor models.

http://sirdurnich.edu

H i g h P Iation of the Electronic Stru Paul F. Batcho

New York University, and Howard Hughea Medid Institute 31 Wnshingkm Place, Room 1021

New York, NY 10003 paul8biomath.nyu.edu

partment of Chemistry and Courant Institute of

.

The material properties of molecular structures are governed by a atmngly nonlinear functional of the electron density and its d a t e d wave function. 'ILpically, the problem is modeled by a nonlinear partial-integral differential system of quationa under a constrained minimization statement. In this talk we will briefly review the electronic structure problem and traditional computational methods for modeling moiecular properties. We then introduce the -t advanced in applying modern spectral methods to the dculation of molecular properties, Phys. Rev. A Vol57,4246,1998 and Phys. Rev. E Voi 61, N0.6, June Moo. Challenges associated with the approximation of wan functions, high precision calculations, and application to mOlecUtar structured will be addrea8ed. Focus will be given to density functional theory-, past, present, and future directions of study and application.

v

ndariea in Fluid Flows, An Experimental Study Jun Zhang

Applied Math Laboratory, &urant fnstituta, .. junBdms.nyu.edu

Modeling DNA-molecules Heiko yon der Mosel

University of Bonn, Dept. of Matthematics heikoQmath.uni-bonn.da

Several different physical system, for example supercoiled DNA molecules, have been m-fully modeled by a framed cum or Line embedded In three dimension8 with an d a t e d energy that h to k minimized mb&t to the cvnstr.int that the c u m not pass through itself. Fbr dosed curvea the knot type may therefore be spedfled a priori, and minima of the energy often appear to involve @OM of self-contact, that is, @OM in which poinb that are distant along the c u m are close In space. While this phenomenon of mlf-wntact is familiar t h u g h every day experience with string, rope and wire, the idea Ls surprislngly difscult to deflne in a way that b simultaneously physically reawnable, mathematically predw and andyticslly tnctable. Here we uo the notion of global curvature of a space curve in a new formulation of the eeleltconket constraint, and exploit our formulation to derive existence results for minimizen of a variety of elastic energies de6ned on m. Nonsmooth calculus is used to derive the Euler equations for the variational formulation of the selfcontact constraint. Theae necmsary conditions provide information concerning regularity of the minimizers and the contact forma.

typically dedt with the eRect on flow of fixed boundaries. The interactions of Buid flows with deformable boundariea are oRan overlooked. This elm of p r o b l w a p a ~ the swimming of Bsh, insect 6ight. Rapping of flaa and continental driR, wbere hydrodynamic fom are strong enough to modi the shape or positions of boundaries. In turn, the boundaries exert fom modifying the Buid flow. Two simple experiments rue presented that illustrate such phenomena. The first experiment, a lower-dimensional vccsion of a flag in the wind, atudies a flexible string in a flowing wap Blm. The system exhibita an interesting bi-stability and the motions resemble the unduiations of a swimming Bsh. In the second example, a thermally convective fluid interacts with a lresly Boating plate, providing a simple model of a continent floatins on the Earth's mantla A periodic oscillation is observed and such solution might be related to the quasi-periodic opening and dosing of the Atlantic Ocean.

Computation of Fast Immersed Boundary Motions Ricardo Cortez

"hlane Univdty, Mnthematica Department co&math.tulane.edu

The problem of a bible filament immersed in a rvo-dimensionel Incompmdbie fluid io considered. The flow is arumed to be fast and length d e s mecroscopic so that the Reynolds number of the flow is large. The goal is to compute simultaneously the motion of the fluid snd the Blament, based on the forces that develop dong it. Thesa forced rue assumed to depend on the geometry of the filament. The approach used is a Lagrangian numerical method that combinea regularized vortices and impulse elements. Each type of element has an evoiution equation. Results for the motion of an undulatory swimming creature will be presented.

Stability and inntability mechanisms of spiral waves Bj6m Sandstedo . Department of Mathematics, Ohio State University ~ t e d e . l e o e u . e d u

mating, travelling or modulated waves arise in many chemical and biological system a3 well as in reaction-diffusion equations on unbounded domains that model such systems. The theme of this talk is to illustrate that the existenca and stabiiity pmpertles of such waves can often be captured by the "spatial- dynamics" approach. The idea is to pca the nsetiondiRuslon equation M a d y n m i d system in the spatial variable and to treat the original time variable aa the new spatial variable. Aa an application of thia appro&, I will dkuaa instubility mechanism of spiral waves such M spiral-wave breakup and the transition to meandering or drifting spinla.

'Ikaveling waves of excitation in a 2-D network of spiking neurons Wemer M. Kistler

S w i ~ Federal Institute of Technology Lausanne Wemer.KistlerQepfl.ch

We study a t d m e n s i o n a l system of spiking neurons with local interactions depending on distance. Thb kind of sptem exhibits a rich repertoire of collective excitations such m traveling waves, expanding rings, and r o t a t b spirds. We describe the underlying dynamics by masns of a continuum description that does not require spatial or temporal averaging. Using this description, propagation velocities and dispersion relations for soUtary waves and periodic wave trains can be calculated analytically. We show that the stability properties of aolitary waves and wave traina are dominated by form instdilities which rue genuine to the two-dimensional nature of the system. We illustrate the analytic reaultn by parallel- computer simulations of s network of loe neurons.

http://paul8biomath.nyu.eduhttp://junBdms.nyu.eduhttp://cvnstr.inthttp://co&math.tulane.edu

Electrical Wave Propagation In the Heart: The Dynamics of Scroll Waver in Anisotropic Excitable Media

Sima SeteJreahgar Caltech, Department of Applied Mathematics

simasr8ama.caltech.edu

There exists growing experimental evidence that fatal cardiac arrhythmias can be aasociated with the formation and subeequent breakup of spiral waves of electrical activity. The goal of b d c and appliad research in this field is a better understanding of the mechanisms underlying the breakdown of coherent wave activity in the heatt, leading to improved schemes for prediction, control and treatment of cardiac arrhythmias. In this work, motivated by the rotation of fiber direction through the thickness of cardiac tissue, we consider the asymptotic theory for the dynamics of scroll waves in the presence of rotating anisotropy (rotation of the fast axis of diffusion). For the case of a stfaight scroll fllament, we have derived a phase equation which is given by Burgers' equation with Coning due to the flber rotation rate. BY considerinn a s d d case of nonconstant flber rotation rate. we have found an exact solution which

.

of the flow. We introducs in a rigorous way the quantity measuring the rate of combustion, which we call the bulk burning rate. This quantity Is well-defined in any burning regime, even if the travelling wave solutions do not exist. For varioue typca of reaction %on-linearities (KPP, ingnition, Arrhenius) we show that there is a class of the flows, which we call percolating, that are very effective in speeding up the reaction. These flows are characterized by the presence of the long tubes of streamlines connecting the burned and unburned material, and in particular include shear flows in a direction perpendicular to the front. In such flows, we prove linear (the strongest possible) lower bound on the burning rate with respcct to the intensity of the flow. The constant of proportionality in the lowcr bound depends explicitly on the scaling proparties of the flow. On the other hand, the flows with closed streamlines, such as cellular, are shown to produce weaker burning enhancemcnt. For such Rows, we show lower bound on the burning rate which behaves like All6 for the large amplitude A of the flow. If time permits, we may also discuss other intsresting phenomena, such as possible quenching of the flame by fluid motion. This is a joiM work with Peter Constantin and Lenya Ryzhik.

& have a flnke-time twist singularity for eerts in initial conditions. This is the Brst analytical result pointing to the deatabilizing effect of cardiac tissue stNCtUre on scroll dynamics. We extend these resub to include the coupling of filament motion to the dynamics of the phase.

Self-starting heat engines as dynamical critical systems. Eric Smith

Los A b o s National Laboratory, and Applied Research Laboratories: Thc University of Texas at Austin

A Strong Comparison Result for Quasilinear Equations in Annular Bounded Domaina and Applications

Rmcesca De Llo Dept. of Mathematics , University of 'Ibrino (Italy)

dalbWm.unito.it

We study "generalized" Dirichlet problems for quasilinear elliptic and parabolic equations. The main aim is to obtain for such equations a Strong Comparison Result, namely a Msdmum Prindple type result between disoontinuow viscoSity s u b and SIJpeRWhtiOM and to ahow we have existence, uniqueness and stability of continuous VisCosity solutions. The approach used allow us to consider ae0 "aingular" equations in particular the geometric equations arising In the d l e d level set approach for dehing the motbn of hypersurfaced with different typm of normal velocities. We are able to provide a level set approach for equations set In bounded domains with Dirichlet boundary conditions.

ITalks on Tuesday May 30, 20001

Bulk Burning Rate in Passive-bctive Diffusion Alex Kiselev

Department of Mathematics, University of Chicago k&3levBmath.uchicago.edu

desmith@arlut.ut.edu

Thermoacoustic engines are especially interesting self-starting heat engines, becaw realizable models have near ideal-gas Simplicity, and because their basic dynamics admits a reversible idealized limit. Self starting hss the character of a rwcond-order phase transition, except that an explicitly dynamical order is formed by breaking of time-translation symmetry. It is shown here that this dynamical transition can be understood as effectively the freezing of thermal noise into a classical background. With a new assumption about how to treat variabletemperature, adiabatic evolution, the dynamical d d p t i o n can be mapped into one in apparent thermal equilibrium, where the phase transition can be found by conventional equilibrium methods. A consequence supporting the validity of the new assumption is that it leads to a very basic proof of Carnot's theorem M a classical current conservation law, requiring only flnite temperature, broken symmetry, and reversibility.

Moment and Almost Sure Lyapunov Exponents for the Solution of the Parabolic Anderson Problem

Leonid Kordov IAS, School of Mathematics

koralov@math.ias.edu

Consider the Stochastic Partial DBerential Quation ut = KAU + { ( t , z)u, t 2 0, z E 2" .

The potential is assumed to be Gaussian white noise in time, stationary in space. We obtain tho asymp I totics of the almost sure Lyapunov exponent for the solution as K + 0. We also study the dcpendcna of the moment Lyapunw exponents upon the diffusion constant K. This is joint work with R. Carmona and S. Molchanov.

Propagation of thin fronta in moving fluids arises in many situations in physicu and engineering. Consider a mixture of reactants interacting in a region that may have a rather complicated spatial structure but Is thin across. The reaction front m o w towards the unburned reactants leaving behind the burned ones. When the reactants am mixed by an ambient fluid then the burning rate may be enhanced. The physical reason for this observed speed-up is believed to be that fluid advection tends to increase the area available for reaction. We study a well-established model of premixed fluid combustion: the reaction- diffusion equation with passive advection. The main question we a d d m is how the burning p- is influenced by the motion of the fluid, in particular how the burning enhancement depends on the geometry

-

. . . . ... . . . . . . . ~~ . _ _ .. ~ ". . . . . , . . , I. . . . . . . , ~~~- ~.

http://simasr8ama.caltech.eduhttp://k&3levBmath.uchicago.edumailto:desmith@arlut.ut.edumailto:koralov@math.ias.edu

For Solutions of Convection-Diffusion Equation with a Fbndom Drift To- Komorowekl

Institute of Mathematics, Polish Academy of sdences, Warsaw and Institute of Mathematics, UMCS, Lubli

komoro~golcmumcs.lublin.pl *

We consider the asymptotic behavior of the aolutiom of scaled convection-diffusion equations ~ Y ( G z ) 1 ~ & 4 , 4 + W'(t /~ ' ,4E) .Bs~( t , 4

t with the initial condition ~ ( 0 , t ) = w(z) M the parameter E 1 0. Under the sssumptions that IC > 0 and V(t, I). (t, I) E R" is a d-dimemional, stationary, zero maan, incompressible, Gaussibn random Bald, Markovian and mixing in t wa show that the law8 of us@, .), t 2 0 considered in an appropriate functional space converge weakly, as E 1 0, to a delta type measure concentrated on a solution of a heat equation with constant d c i e n t s .

Exploring Transitions in Tu0 Fluid Interface Syatems Ita1 a h e n

University of Chicago Phydca DepL icohenQdhy.uchi

Understanding and controlling how a liquid interface its topology from being singly connected to being multiply cdnnected (M is the cam with a drop dripping Irom a faucet) or from being bounded to being unbounded in a particular direction (as b the caw in tbe seleaivs withdrawal problem) b crucial for the control of many manufacturing pro- including the meation of emulsiom and monodispersed sprays. Furthermore, the mathematics describing these typea of topological changea b not understood very wll. In thb talk I will discucls two brAutiful phenomena which explore and shed light on these issues. First I will discus experiments on the two fluid drop snapoff problem where fluid is dripped through an outside liquid medium which b viscous. I d l then go on to d m the Selective withdrawal problem. Here we lower a straw so that ib M c e rests a b a wateroil interface. We then withdraw the oil through the straw. By changing the rats of withdrawal wo control a transition between haviw only oil being withdrawn and having water being entrained along with the oil.

Some Results on the Evolution of Thin Non-Newtonian Jeb Manx, A. Fontekm

Univemidad Rey Juan Carlos, ESCET, Area de Matematic8 Aplicada fonteh8anad8to.enc%t.urjc.es

The evolution of thin viscous jets of polymeric fluids pments significant differences with respect to the evolution of ordinary Newtonian fluids. The most dgnifimt, from the point of view of applications, is the fact that break-up of the jet is avoided for long periods of time, We will show that this fact follows from the equation8 that model the flow and diseusa some other properties. These properties will also depend crucially on the Non-Newtonian model under consideration.

New analytic methods for PDEs In the complex domain Ovidlu C a t i n

Rutgem University, Mathematics Dept. costinc0math.rutgen.edu

I will discuM new technique3 to show existence and uniqueness of sohtions of aome classes of nonlinear PDEs in the complex domsin, and to study the formation of ainingularitiea (in certdn regio~) . Examplea arising in the context of Hele-Shaw dynamics will be disc&.

[Talks on Wednesday May 31, 20001

Homoclinic orbit8 for near-integrable PDEa Chongchun Zeng

Eengchamathl.nyu.edu CoUMt IMtitUh

In reant yeas, there have been extensive eudies on the existence of homodinic orbits for dissipative near integrable PDIh, which are closely related to chaos In thb nork, we consider and a perturbed sine-Gordon equation

utt - a'u, - sinu = e(ulrt + I(:,u, ue)) and a perturbed nonlinear Schr6dinger equation

iw = %s + 2(ua - W')U + ie(utS - au - p) for u even and periodic in r. We prove the existence of homocllnfc orbits for these PDEe.

(The result for the sineGordon equation ia a joint work with Jalal Shatah, Courant Institute.)

Chaotic dynamic6 from a probabilistic viewpoint Marcdo Vlum

lMPA - Rio de Janeiro vianatbiipabr

An overview of some recent results, by sewid authors, pointing towards a global theory of dynamical systems and their attracton, in a gwmetric and probabilistic h w o r k .

Eddy gmwth criteria via a Melnikov approach Sanjeem BalasuriyP

Brown U n i d t y s&eva&fm. bmwn.edu

The dilfudve effect on bamtropic models of d eddieu b addregsed, using the MeK~kov method from dynramical rystema. Simple geometric criteria are obtdned, which identify whether a given eddy grow 01 d h out, under a di&lsive (and fordng) perturbation on a potential vorticity c o m i n g Row. Qualitatively, the following are shown to be featurea conducive to eddy growth (and thereby stability in a spedBc awe): (i) large mdiua of curvature of the eddy boundary, (U) potentld vortlcity contoun more tlghtly p.cked within the eddy than outside, (iii) acute eddy pinch-angle, (iv) small potential vorticity padient ruxw the eddy boundary, and (v) meridional wind forcing which increases in the northward

http://costinc0math.rutgen.eduhttp://Eengchamathl.nyu.eduhttp://bmwn.edu

H,, , the locations of the vortices being speciRed. We will focus on a wsult in which a vorticity-measure is deRned for energy-minimirars, and shown to converge to a limiting density of vortices. This density is uniform in a subset of n determined by a free-boundary problem depending on h,.

Macroscopic models for Ostwald Ripening Barbara Niethammer

Univemity OS Bonn, Department of Applied Mathematics Barbara.NiethammerBiam.uni-bonn.de

Ostwald ripening is a fundamental process in the aging of materials, where many particles of one material embedded in another material undergo coarsening to reduce the total interfacial area. For applications it is of considerable interest to predict the coarsening rate of the material, e.g. the increase of the mean size of the particles.

In the regime of small volume fraction of particles Lifshitz, Slyom and Wagner (LSW) heuristically derived a noniocal conservation law for the particle size distribution. Experiments, however, only partly confirm predictions based on this model. It is therefore natural to ask in which regime the LSW-model is strictly valid.

In this talk we will show how a rigorous mathematical analysis, based on the homogenization of a free boundary problem, can clearly identify the natural range of validity of the LSW-model. The difeculty b that one particle interacts with a large number of other particles, and hence it is important to understand the effective range of particle interactions. If acmening effects are not negligible wc obtain an inhomogeneous extension of the LSW-model. Quite surprisingly, the analysis can also be extendcd to two-dimensional system, even though the formal argument by LSW does not apply here.

direction. The Melnikov approach also suggests how tendrib (Rlaments) could be formed through the breaking of the eddy boundary, M a diffusion-driven advective prooess.

Thia is joint work with Chris Jon-.

Harnack type estimates for cooperative elliptic system Boyan Stakw

University of Paria 10 s i rMann. jwieu . f r

We study elliptic systems of the type &ut +CII (Z)UI+. . .+~~~(Z)U. = It(%).

( L n s + ~ni(z)ut + + Gn(z)h = In(z), b a + ~1i(z)ui + - + Q&)u, = fib) ... where Ld are secondsrder elliptic operators in non-diwrgence form, with bounded measurable coefficients. The functions c j are supposed to be bounded measurable and cj 2 0 for i + j (cooperative system).

We axtend the classical Aiexandrov-Bakelman-Pud and Harnack-Kryh-Safonov estimates for seaiar equations to this trpe of systems, and describe the particularities due to the coupling in the system. We give a number of applications -maximum principles for systems, elliptic estimates for higher order equations (polyharmonic equation).

.

This is ajoint work with Jerome Busca.

Minimal Surfaceg and Topological Defect8 Randall D. K d e n

University of Pennsylvania, Department of Physics k Astronomy kamienOphysics.upenn.edu

Large twistangle grain boundarb in layered structurer, are often described by Scherk's Rrst surface whereas mall twbtangle grain boundaries are usually described in terma of an array of m e w dislocationa I will discusll this and other minimal surfam and will show that there is no easential distinctbn between minimal surface and topological defect dencriptlons and that, in particular, their comparative energetics depends crucially on the core structure of their acrew-dislocatbn topological defects.

Vorticity for the Gineburg-Landau equation of superconductivity Sylvia Serfaty

Ecole Normale Superieure de Cachan clerfatyOcmla.en&an.fr

The Ginzburg-Landau functional 1

~ ( u , A ) = 1 I V A ~ + lcuru - LIZ + s(i - IUP)~ is used by physicists to model superconductors in an applied magnetic Reld. There has been a lot of mathematical work related to this functional in 2 dimensions, particularly on the simpliRed functional F(u) = Jn lVula + &(1 - 1 ~ 1 ~ ) ~ since the work of Bethuel-BrezkHelein. We report on recent reaulta (partly joint work with Etienne Sandier) on the full functional in the limit E 4 0. Minimizing solutions are characterized by the existence of vortices (zeros of u) for values of the applied Reid h, between two critical values H,, and H-. Results include an sPymptotic expansion of He, as c -I 0, the coexistence of branches of stable solutions with n vortices, for n arbitrary integer, and hu in a large interval around

Acoustic pulse shaping and localization in a random fractal Knut S z h

University of Utah, Department of Mathematies solnaQ math.utah.edu

We consider wave propagation in multiseale random media. Our objective is to characterize how a pulse propagating through the random medium is affected by the medium Ructuations.

M u l t W or f r d random media are used to model for instance the heterogeneous earth and the turbulent atmosphere. For a claw of fractal random media defined in terms of fractional Brownian motion we show bou an acoustic wave pulse interacts with the medium Ructuations. The modification in the pulse shape depends on the roughnwa of the medium and can be described in a detcnninwtic way when the p u b is obscrved at ita mndom arrival time. We show that for very rough mdia the wave is local id to a surface layer.

Nonlinear Diffusion in Computer Vision Sibel Tad

Middle East Technical University, Department of Enginwring Sciences stariQmetu.edu.tr

Computer vision deals with detection of singularitiea from discrete imagw of the s e n d world. Sin- gularities of image gray level arise due to existenm of shape boundaries. On thc other hand cingularities of the shape boundary are the locations at which perceptual information Is conmntrrtcd. I will prqent an approach for recovery of perceptual information from images based on invariant s p m and non-iincar reaction-diffusion equations.

http://Barbara.NiethammerBiam.uni-bonn.dehttp://sirMann.jwieu.frhttp://kamienOphysics.upenn.eduhttp://clerfatyOcmla.en&an.frhttp://math.utah.edu

ltigrid for Imaging ctrlcal Conductihty and Permittivity at Low Fhquency

Lillana Borcea

borCG8OCMtIl.rica.edU .b ty, Computational and Applied Mathematics

, We propose a nonlinear multigrid approach for imaging the electrical mnductivity and permittivity of a body R,’given partial, usually noisy knowledge of the Neumann to Dirichlet map at the boundary. The algorithm b a n&ed iteration, where the Image is mnstructed on a sequence of grids in R, starting from the marse grid and advancing t o w ~ d s the fin& one. We show variou numerid exmplm that demonstrate the elktiveness and robutneaa of the algorithm and prove local convergence. L

A Parallel Distributed Climate Model for Precipitation Mrau MOlller

L a m Berkeley National hboratory AmuellerBlbl.gov

Climate reseafeh provides one of the most challenging trska in high performance computing. The pre- sented model ’Masoscale Atmospheric Simulation’ maintained by the Earth Science Division at Lawrence Berkeley National Laboratory focuses on the simulation of precipitation in the mssoscala. Differant, seg amtely mmputed mmponants lllaa advection, mii, radiation and doud mimucloud models mntribute to this simulation. The simulated box in the atmosphere and the governing equatiom are discretized on a regular threedimenslond Ardtawr Egrid which b applied to all modules of the model. Ail mmponenta are calculated consecutively by l d y coupling. Simulated values are used M initial dues fir the suc- d i n g module. The paralklization strategy is a twdimensional domain decomposition of the regular grid in the horizontal directions. Due to the calculation schame of the discretized transport equations using difference stars the CPUs working on dilferent domdns need to exchange data. By MOM of porta- bility to platfomrcl with d tsc turecr using mmmon or distributed a d d m spacea the message passing standard MPI has been used. With an implementation wing distniuted data approach the paralielizsr tion has been reallzed on the CRAY T3E at NERSC. Lawrenca Berkeby National Laboratory. Achieved results of the p a r d l e W i n will be shorn.

ITalks on Thursday June 1, 20001 I 1 1

Parameter Identiflcation in Nonlinear Abstract Cauchy Problems

Ruben D. Spies Institute de Matamdtica Aplicada del Litoral, MAL,

&nsejo Nacional de InvsstigsdolKs Cientfflcm y TBcnic~, CONICET, Universidad Nacional del Litoral, UNL,

Centro Regional de InvestigDddn y Decwrollo, CERIDE, Sants Fe, Argentina

* Using Quasilinearisation

~Ph8peIIlM.Ud.dU.U

An approach to quasliinearization for parameterldentiflcation in nonlinear abstract Cauchy problems in which the parameter appears in the nonlinear term, h prssented. This approach has two main advantages over the ddcal one: it is more intuitive and the derivation of the algorithm is done without need of the aensitivity equations on which classical quasilinearlzation b based upon. Sufficient mnditions for the mnvergenm of the algorithm are derived in ten^ of the regularity of the solutions with respect to the parameters. A mmparison with the standard approach is presented and an application example is shown in which the non-physical parametem in a mathematical model for Shape Memory Alloys are estimated.

Kinetics of materials with wiggly energiea and related averaging problems Govind Menon

Brown University, Division of Applied Mathematics govind&fm.brown.edu

1’11 dkum the dynamka of gradient syatem~ with ‘Mgglyenergiea”. These ODE are directly motivated by -nta on phaw tranaitiona in &ape memory alloyq and are mars? models for the dynamica of system with microstructure. The mathematical problem is to average/homogeniZe these dynamical systems and efficiently capture the dect of miaoatructure. Mostly, I’ll talk about a simple model lor a m s s s k partlde sliding down a rvngh d a c e , and ita surprisingly complex structure.

Moving Interface Problems, Numerical Method8 and Applications. Hongbl Zhao

Department of Mathematics, university of California, k v i . zhaoCtmath.uci.edu

Moving intecfaca and frm boundary p r o b k are of great interest in many pbysical and mathematical study. I will present a few interesting ammples in fluids, mataials and geometric flows, such M, two phase flow, aystal gmwth, surface Uusion, mean curvature and Oausdan curvature flows. The main part of my talk will f m on the numerical challenges in the numerid mmputation of moving interface and free boundary problema In particular I will diiw in noma details about the level set method and ita variational formulation. In the end I will illustrate mme recent applications of the level set method to shape reconstruction and deformation in computer graphics.

http://borCG8OCMtIl.rica.edUhttp://AmuellerBlbl.govhttp://govind&fm.brown.eduhttp://zhaoCtmath.uci.edu

Attraction and Diffusion with Hydrodynamic Interaction Michael Brenner

Mathematics, Massachusetts Institute of Technology brenn4math.mit.edu

There are many open Important questions involving the the interactions betweea very wall objects In a viscous Ruid. AIter summarizing some general questions and applications that I nm thinking about, I will discuss work on two examples. The b t (joint work with Todd Squires) is s theory for recent experiments observing "like charge attraction" between coiloidal spheres in conlined geometries. The second (joint work with Boris Shralman and Peter Mucha) lnvolvea our attempts to mre out from Erst principles) the effective equations describing a monodisperse suspension of particles at low volume fraction.

Biologically Inspired Learning Algorithms Daniel D. Lee

Bell Laboratories, Lucent Technologies ddleeelucent.com

Information proceasing capabilities of artiRdal system presently lack the robustness and rich com- plexity found In biological system. Endowing artificial system with the ability to adapt to changing conditions requires algorithm8 that can rapidly leanr from examples. I will demonstrate the application of one such learning algorithm on an Inexpensive robot dog mnstructed to perform simple sensorimotor taclks. The robot learns to track a particular object by discovering the d i t visual and auditory cues unique to that object. The system usea a convolutional neural network to combine color, Luminance, mo- tion, and auditory information. The weights of the networks ace adjusted wing feedback from a teacher to reRect the reliability of the various Input channels in the surrounding environment. I will also discuss how unsupervised learning can d w v e r features in data without external interaction. An unsupervised algorithm based upon nonnegative matrix factorization is able to automatically learn the Werent parts of objects. Such a p8rta-W representation of data b crucial for robust object recognition.

Cellular signalling: Composing global rignab from elementary events Stephen Coombes

Nonlinear and Complex System Group, Dept of Math Sciences, bughbormgh University S.hmbe5OLboro.ac,uk

The existence of spatial and temporal signalling by calcium is one of the most signiacant findinga of the last decade in the field of intracellular signalling. Caldum is stored intracellularly in the endoplasmic or sarooplasmlc reticulum at 2-3 orden of magnitude greater than its concentration in the cytosol and is released by a nonlinear feedback process referred to en calcium-induced +lcium release (CICR). This m-m for generaling oscillations (or puffi) in the density of cytosolic free calcium h b e l i e d to underlie the waves that propagak, en intra and intereallulu wavea over distance PI large M lmm. A n i m k phenomenon is thought to occur in the dendritic tree of neural cells. Voltage s p i b in dendritic s p i n s h e d electrically connectad to a dendritic tree can induce cunentr In neighboring spines via the diffusive spread of witage along the tree. If the induced current in neighboring spines is suWcientiy large a spike or action potential may be generated leading to a travelling wave. We diaeuss and analyse modeis that acq capable of composing giobd sign& (travelling waves) from elementary events (puffi or spikes) for both the above biological rystem.

stability of the asynchronous state in networka of atrongly coupled oscillators Carl van VreeswUk

Gatsby Computational Neuroscience Unit, University College London earl@gntsby.ucl.ac.uk

Over the last twenty yeam networks of weakly coupled limit-cycle oscillators have teen studied ex- tensively. Because the coupling is weak in these systems, it does not significantly drive the units away fmm their limit-cycle. As a result, the state of each unit is, to leading order, completely denwibed by Its position on the limit-cycle (its phase) and the Interaction betwecn pain oscillators can be described by function that depends mlely on the phase difference betweeh these units.

A crucial step in arriving at the phaseeoupled model is neglecting perturbations orthogonal lo the limit-cycie. These can indeed be neglected for weak coupling. However this is not the m e for stronger coupling. In the latter cam the shape of the l i t -cycle is deformed in a manner that depends on the state of the network and to study the stability of this state one also needs to take into account possible destabilizations of the system through modes that are orthogonal to the limit-cycle.

The extremely bmad range of phenomena that can be encountered in such systems makes It very unlikely that a suffidently simple general mathematical framework exists for the general study of such system. A more fruitful appmach is (at least for the time being) the study of spocial cases.

I will explain the stability analysis of the asynchronous state in a liomogencous network of all-ball coupled osclllaton and show an example in which this state Is d e s t a b i l i through an orthogonal mode. The possible relevance of this destabilization for newbiology will also be discussed. Finally I will briefly touch upon extendons of tbe analysis to inhomogeneous networks, networks with noise and the analysis of the Nly synchmnized state.

Approximate controllability and homogenization of a semilinear elliptic problem. Axel 0-

Department of Mathematical Engineering, Unlversidad de Chile. axosaesBdim.uchiIe.cl

We study the approximate controllability of the semilinear clliptic equation -div (A'Vy') + f(v') = 1-Y'

in a domain fl of RN with Dirichlet boundary conditions. Here F is an homogcnization parameter which converges to zero, A' is an uniformly elliptic coefficient matrix of class 0, f' is a continuous globally Llpschitz non linearity and w is a control rubdomain strictly included in fl. Given a N - 1 dimensional manifold S into fl non intersecting w, it is known that, under reasonable topoiogical restrictions on S, for a given a > 0 and y1 there exists a control d such that the trace on S of y' lies in a ball of center 81 and radius a which is constructed explicitly. We study the behavior of tf as the homogenization parameter tends to zero. We prove that there exlsts a limit control resulting as the limit of Rxed points for each controllabiiity problem In E. We link this limit control with the corresponding homogenized problem.

This is Hint work with C. Con= (U. de Chile) and J. Saint Jean Paulin (U. de Metr, France).

http://brenn4math.mit.eduhttp://ddleeelucent.commailto:earl@gntsby.ucl.ac.uk

. . DNSbaaed predictive control of turbulence:

an optimal benchmark for feedback algorithm

Thomas FL k l e y Department of MAE, UC San Diego, La Jolla, CA 92093, USA

bewleyQucsd.edu

Direct numerkal simulations (DNS) and optimal control theory are uaed In a predictive control netting to determine controls that effectively d u m the turbulent kinetlc energy and drag of a turbulent flow in a plane channel a t Re, = 100 and Re, = 180. Wall transpiration (unsteady blowing/auction) with zero net mass flux is used as the control. The algorithm wed for the control optimization b based solely on the control objective and the nonlineat partial diR8rentl.l equation governing the Row, with no ad hoc assumptions beyond that of a finite prediction horizon, T, over which the control is optimized.

Flow relaminuization, accompanied by a drag reduction of over 50 p r o d , is obtained in some of the control eases with the predictive control approach in direct numericel simulations of subcritical turbulent channel flows. Such performance far exceeds what has been obtained to date in similar flows (wing thm type of actuation) via adaptive stmtegk such as neural mtworka, intuition-bad strategies such as opposition control, and the d l e d "suboptimal" strategies, which involve optimizations over a vanishingly small prediction horizon P 0. To achieve flow relaminlrization in the predictive control approach, it is shown that it is naasMy to optimize the controls over a sumdently long prediction horizon T+ 2 25. Implieatiom of thia result are discussed.

The predictive control algorithm required full M e l d and is computationally expensive, involving iterative direct numerical simulations. It b, therefore, impcnsible to implement this algorithm directly in a practical setting, However, these calculations allow us to quantify the best possible system performance givan a certain classof flow actuation and to qualib how optimized controls conelate with the near-wall coherent structures believed to dominate the pmcesr of turbulence production in wall-bounded flows. Further, variow approadras have been p r o p o d to distill practical feedback schemes from the predictive control approach without the suboptimal apprcadmatlon, which b shown in the present work to restrict severely the effectiveness of the d t h g control algorithm. The present work thus representa a further step towards the determination of optimally effective yet implementable control strategien for the mitigation or enhan-t of the consequential effecta of turbulence.

..

Passive scalar intermittency and the moment problem. Jared C. Bronski

University of Illinois Urbana Champaign, Mathematics jared0math.uiuc.edu

It is a wet1 documented, experimental fact that while single point velocity measurements in turbulent fluids admit nearly Gaussian statistics, measurements of other quantities including velocity increments, vorticity, pressure, and passively transported quantities admit strongly non-Gaussian statistics. There has been an in- and important phenomenological effort attempting to explsin thia behavior in the context of a passive scalar using a variety of doatre appmdmationa and other d h o c approximations. Even in the context of a passive d a r , the question of inherited statistics is extremely difficult, requiring the consideration of partial differential equatiom with variable coefficients, in very large dimension.

In this Iceturn, we study the model introduced by Majda which concern a passive scalar decaying in the presence of a rapidly fluctuating, Gaussian linear shear profile. In joint work with Richard M. McLaughlin (UNC - Chapel Hill), we present the first explicit construction of the limiting asymptotica for the momenta of the normalized d a r , and UM this information to rigorously deduce the pdf tail. This elementary 6eld theory shows the pdf tail ranging from Gaussian through stretched exponential a parameter is varied, and we additionally obtain an explicit relatlon between the tail of the d a r and the tail of the scalar gradient which explkltly demonstmtea how derivatives may become more intermittent.

Weak turbulence theory: development and novel applications. Yurl V. LMY

Department of Mathematical Sdencea, Ranswlaer Polytechnic Institute IvoVyBrpi.edu

What semiconductar laser theory, Rber optica, surfacnwater waves and acoustic waved have in common? Although these system are lleemingly dinconnected and have quite dfferent physical nature, they can be viewed M complex system composed out of interacting particles or waves. There is a general theoretical framework for their statistid description, called weak turbulence theory. Om can obtain a c l o d equation describing the tima evolution of such system, called kinetic equation. I will explain what el- of stationary solutions klnetic equation has, and how understanding of rurfacs water waves can lead to better design of semiconductor lasers.

ITalks on Friday June 2, 20001 1 I I

Kinetic Equilibration Ratea for Granular Media Robert J. McCann

University of Toronto Department of Mathematics meeannQmath.toronta.edu

'

This joint work with JoSe Carillo and Cedric Villani provides an algebraic decay rate bounding the time reguired for velocities to equilibrate in a spatidy homogeneous flow-through model representing the continuum b i t of a gaa of partkb interacting through slightly inelastic collisions. The rate is obtained by reformulating the dynamical problem M the gradient flow of a convex energy on an infinitdimensional Riemannian manifold. .An &tract theory is h i o p e d for gradient Bows which shows how degenerate convexity (or even non-convexlty) - if uniformly controlled - will quantify contractivity of the flow.

Front propagation phenomena in periodic media F. Hamel

CNRSUniversit6 Paris VI hamelQann.jussieu. fr

Advectiondiffusion-reaction e q u a t i o ~ for p d v e quantities can give rise to various propagation phe- nomena: while travelling fronts propagating in a given direction with constant speed and constant shape appear naturally in media which are invariant in this direction of propagation, pulsating (or periodic) travelling front8 moving with an effective speed and periodic shape appear in media which are periodic through the underlying velocity field, the d i h i o n matrix or through the geometry. Such periodic d e mains include for instance the in6nite cylinders with straight or oscillating boundaries, the whole space with periodic holes.

I will especially talk on pulsating travelling h n t s fir semi-linear elliptic equations arising either in combustion models for flames or in models of dynamics of population like Fisher-KPP type equations. Of particular interest is the determination of the effective speed of propagation. In some casea where there axista a bounded from below half-line of possible speeds, the minimal speed can be given by variational formulas in tmna of related elgenvalw problem in the cell of periodicity of the domain.

"his talk is based on joint works with R. Berestyckl and N. Nadirsshvili.

http://bewleyQucsd.eduhttp://jared0math.uiuc.eduhttp://IvoVyBrpi.eduhttp://meeannQmath.toronta.edu

A global approach for the atudy of blow-up in nonlinear heat equations Hatem Laeg

Courant Institute and CNRS zaagQdms.nyu.edu

Blow-up in nonlinear heat equations captures features common to a whole range of blow-up problems arising in physics and geometry (surfcur, dlffusion, &mot axis...) However, nonlinear heat equations are simple enough to be tractable in rigorous mathematid terms. We will preaent a Liouville theorem for entire solutions (defined for all time and cpace) of a nonlinear parabolic equation. This theorem allows us to adopt a new approach in the study of blow-up lor a related nonlinear heat equation, based on global estimatps. In particular, we obtain blow-up estimates, uniform with reapect to initial data and the blow-up point. This way, we prove the stability of the (generic) blow-up profile for this equation.

Entropy dissipation method for parabolic equatiotls Jose A. Carrillo

University of Texas at Austin; on leave from Universidad de Cranada, Spain. carciQmath.utexacl.edu,dlloBugr.es

The mnvergence of the L'-solution of the porous medium equation to the Barenblatt pro6le hes been known since the works of Medman and Kamin in the late 70's. Havever, the rata of mnvergence to equilibrium has not been a d d d until recently in the natural space of integrable functlons. The entropy dissipation method originally introduced in kinetic theory of gases allow us to treat more general nonlinear parabolic equations containing special CB(KS the porous medium, the k t diffusion and homogeneous granular media kinetic eg~ations. The method is based on the evolution of a Liapunov functional for these dynamica 1 sysrs3nrr. "hb presentation is a summary of seved works in mllabontion with R McCann, P. A. Markowich, 0. 'hacanit A. Unterreiter and C. VUani.

Control on the distribution function of particles in a Stokes flow Pierre-Emmanuel Jabln

&le Normale Sup&ieure, Parla jabinBdmaena.fr

This is a joint work with F. Otto where we lnwtigate the dynamics of rigid particles in interaction in a fluid. We neglect both the lnertia of the particles and the fluid. We show that, in a dilute regime, if the number of particles is large enough, they cannot get significantly closer than what they were at the initial time. This result b essential if one wants to rigourously derive a model when the number of particles becomes intinite.

Global Well-posednslre for dispersive equations with rough data Gigliola Staftllani

Department of Mathemati- Stanford University gigliola@math&anford.edu

In recent yeam there has been a aeries of fundamental resulb on well-posedaess for evolution problem of dispersive type (like Schriidiger, KdV, KP and also wave equations) With rough initial data. In general these resulb are local in time. One c l m i d way of extending a loca result to a global one is to prove an a priori uniform bound for the some Sobolev norm of the solution exploitihg the dasslcal conservation laws of tho equation. Thii norm usually is either La or H1 and it is called energy norm. About two years ago Bourgain introduced a new general strategy to attack the problem of global well-poscdness. Hi method

consiata on splitting the solution into a part involving low Fourier modes and a part involving high ones, and recognizing that the "low-mode" part is in the energy space and the "high-niode" part has a small contribution in the dynamics. In particular Bowgain p r o d that for a two dimensional cubic Sddinger equation With initial data a bit less regular than the one in the energy space, the solution exists globally.

Later Bourgain method was appl i i sucrpsSruUy by several authors to the study of other dispersive

Recently, in mllaboration with J. Colliander, M.Keel, H. Takaoka and T. Tno, we introduced a new method that allowed us to show that in the case of KdV and modified KdV (both periodic and nun periodic) local existence is equivalent to global existence. The method of p m f relies on a new claas of almost conservation laws, de6ned through the Fourier Tkansform, which are particularly useful when the solution lacka smoothness. We believe that our method muld also be applied to system that are not integrable, like lor example the two dimensional cubic Schrcidinger equation treatad by Bourgais. Moreover the consemtion laws we introduce give some new insights into the dynamics behind the the La norm.

eqU&bM.

On the t1 Well-pasedness of Solutions to Hyperbolic Systems of Connewation Laws with Large BV Data

Komtautina M v h Northwestern University, Department of Mathematics.

trivisaQmath.nwu.edu

We consider the Cauchy problem for the strictly hyperbolic system of n conservation laws in one space dimension: (1) W(z, t ) + B.F(U(z, t ) ) = 0, (2) U(z.0) = i?(z). Each characteristic fleld is &ssumed to be either linearly degenerate or genuinely nonlinear.

Major progreag in the theory of hyperbolic sysystems of consemtion laws haa been the proof of the stability of solutions to (1)-(2) with initial data of moll total mriation Ill, 121, 131. A significant problem in the field, which remains open, is the establishment of the well-pdness of

solutiotu to (1)-(2) with initial data i? being bounded but possibly laqe. Aa a 6ret step in that direction, we consider as idltial data 0 a small BV perturbation of a fixed

Riemann Problem (U&Ui) whose solution contpins two laqe stable, Lax compressive shocks traveling with dmerent characterlstic spmda A' and Aj.

We prove: 1: The (global) existence of entropy clolution to (1)-(2) with initial data 0 suitably close to the

2: The stability of the solution U under small BV perturbations of its tiitial data. The principal took in the analysis are the wave front tracking algorithm and the notion of an entropy

Riemann data (U&Ui).

functional. ( W ir joint work with M. Liwich.) REFERENCW

[l] A. Bnuo, 0. Cruta., uul B. Pieolll, Well pmdneaa of the Cbuchy probbm br n x n syatenu of mrurzvatbn law.

[Z] A. Brasan, T. P. Wu, and T. Yuy, L1 Stability EsUm.ta br n x n mwe.rmtion lam, A d . Rohmul M c r h And,

[3] T. P. Llu and T. Yaw, LI Stability of sdutkma ol Hyparbolic S-nu of Conservation Lam, Con& Amen MeU*

M n n d r A w . M d h . Soc. to w.

to WPSU.

soc., to .ppu.

.

http://zaagQdms.nyu.eduhttp://jabinBdmaena.frmailto:gigliola@math&anford.eduhttp://trivisaQmath.nwu.edu

Maxwell-Klein-Gordon Equations in lklimensional Minkowski Space Maria Pmrelli

Bronx Community College of the Cit$ Uniwrsity of New York peardliwma.nyu.edu

I study the asymptotic behavior of the d U t b M of massive' coupled Maxwell-Klein-Gordon Relds equations in 4 4 Minkowski space. In particular, I deal with a general version of the system that allows for the presence of charge in the initial data together with am= term in the Klein-Gordon equation. This results in new issues that cannot be handled by standard methods. A covariant Lie derivative operator for the Klein-Gordon field Is introduced to take care of the troublesome tams. In the case of small initial data, I And that the gauge invariant LOD norm of the Klein-Gordon Reld decays uniformly like the free solution and that the decay of the electromagnetic Reld differs from the free one by a logarithmic bss. The proof is based on gauge i n d a n t energy estimates and geometric propertla of the fields equations. Moreaver, I derive for the case of large initial data a weak gauge i n d a n t decay estimate of the solutions of the system which implies that the Klein-Gordon Reld decays to ram in the local L' nom.

%

New High-Resolution Central Schemea for Consenration Law, Hamilton-Jacobi Equations and Related Problems

Alexander Kurganov University of Mirhigan, Department of Mathematics

kurmmath .krumidr .edu

We introduce new Goduaav-type semi-discreta mtnl schemps for hyperbolic systema of conservation laws and Hamilton-Jawbi equations. The schemes are based on the we of more precise infomation about the local speeds of propagation, and can ba viewed as a generdition of the schemes from [2J] and 131.

The main advantages of the proposed central sehemas are the high resolution, due to the smaller amount of the numerical dissipation, and the simplidty. "here am no Riemann solvers and characteristic decomposition involved, and this maken them a universal tool for a wide vsriety of applications. At the same time, the developed schema have an upwind nature, since they respect the directions of wave propagation by measuring the one-sidul bed speeds.

ad the Euler equations of gas dynamics, the Hamiltoo-Jambi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navkr-Stoked equatlona.

(joint work with Sebastian Noelle, Bonn University, and Guergana Petmva, University of Michigan)

[l] A. KUROANOV AND D. LEVY, A thircCoder remi-diamte mkal # & m e for wnamation law and w n v e c t w n - d i m n apuoiiow, SIAM J. sei. Comp., to appear. [2] A. KURCANOV AND E. TADMOR, New high-rwohtion cmbal rehrmer fm nonlinear amservotirm lava and w n v c c t i o n - d i m quatione, J. of Comp. Phys., to appear. I31 A. KURCANOV AND E. TADHOR, New high-nrolution rm*-di#mte rehmru for Hami&on-Jacobi eguotim, J. of Comp. Phys., to appear.

The constructed schemes are applied to various problems,

Evolution Galerkin schemes for multidimensional hyperbolic systems Mbrh Lukd&v&-Medvtawd

Technical University Brno, Czech Republic dc Otto-von-GuericksUniversitiit Magdeburg, Germany Lukacova@fme.vutbr.a

Thb contribution deals with gsnulnely multidimensional numerical schemes for solving hyperbolic systems. In recent years the most commonly used methods for hyperbolic problems were finite volume methods which were bamd on a quad dimensional splitting using one-diiensional Riemann solvers. It ttUM out that In special cases this approach leads to structural deRdencies in the solution.

The d m b to construct A method which takes Into amount all of the infinitely many directioh of waves propagation. The main idea of the evolution G a l e r h methoda is the following: the initial function is evolved by m e a ~ of approximate evolution operatom dong the bicharacteristic cone and then projected onto a finite element space.

The deacribed approach has been fully exploited for the linear hyperbolic syatems such M the wave equation clyatem or the Uaxvau equatbna When the evolution Galerldn method is applbl to mora complicated nonlinear comervation lam, and the solution may contain shocks, a auitable linearization has to be made and the time step b limited ty the resulting apprmximation.

Our purpose is to describe a number of appmximata evolution operators for hyperbolic systems in two space dimensions, in particular for the nonlinear conservation laws such as the Euler equations. Prelim- inary attempta are put on devebpment of new practical numerical algorithms. We present theoretical analysis of the &mea, dicmuw the stability propertiea well as the error estimates and show some results of numerical experimeta and comparisons with other commonly used methods.

The p-t cedearch hm been done in cooperation with 0. Warnecke, Otto-von-Cuerielre-Universitat Magdeburg, and K.W. Morton, Oxford University. Author gratshrliy acknowledgea a support of the DFG Grant No. Wa 633/6-2, the Grant CZ 39001/2201, and the GACR Grant 201/00/0557.

http://peardliwma.nyu.eduhttp://kurmmath.krumidr.edu

IPosters on Sunday May 28, 20001

The random projsetion method for hyperbolic conservation laws with &iff reaction t e r n Welshu B ~ Q and Shi J l n

School of Mathematics. GeorgLa Institute of Technology Atlanta, Georgia, GA 30332. USA wWmath.gatech.edu

In this talk WB propose the random projection msthod for numerical simulations of the hyperbolic conservation laws with stiff source terms arising from chemically reactive flows:

In this problem, the chemical time scales may be orders of magnitude faster than the fluid dynamical time sealos, making the problem numerically stiff. A classic spurious numerical phenomenop, the incor- rect propagation speeds of discontinuities, occurs in underresolved nu