FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES : HYDRODYNAMICAL LIMITS ... › 2018 › 03 ›...

Particles, gases, fluids and the Hilbert’s 6th problemGeneral presentation of the Boltzmann collisional model

The incompressible Navier-Stokes limit of the Boltzmann equationQuantitative study of the convergence towards Incompressible Navier-Stokes

FROM BOLTZMANN TO INCOMPRESSIBLENAVIER-STOKES : HYDRODYNAMICAL LIMITS

AND SPEED OF CONVERGENCE

Marc Briant

Laboratoire MAP5, University Descartes (Paris 5)

Workshop on kinetic and fluid partial differential equationsUniversites Paris Descartes & Paris Diderot, March 7th-9th 2018

Marc Briant From Boltzmann to Incompressible Navier-Stokes

default Particles, gases, fluids and the Hilbert’s 6th problemGeneral presentation of the Boltzmann collisional model


Table of Contents

1 Particles, gases, fluids and the Hilbert’s 6th problem

2 General presentation of the Boltzmann collisional model

3 The incompressible Navier-Stokes limit of the Boltzmannequation

4 Quantitative study of the convergence towards IncompressibleNavier-Stokes




Particles, gases, fluids andthe Hilbert’s 6th problem




Particles systems

Systems under consideration : N bodies of identicalmass moving on the torus Td

Each particle is influenced by external forces and undergoesinteractions with other particles

Newton’s laws give rise to the celebrated N-bodies problem (asystem of 2Nd equations)

dxidt

= vi ,dvidt

= −Fext(xi )−N∑

j=1, j 6=i

∇Φ(xi − xj)

⇒ Already problematic for N > 3 !




Different scales of description

Macroscopic movements : fluid equations : we lookat the dynamics of the global mass ρ, the mean velocity u andthe total energy E of the system considered as a continuousmedium

Several modellings for continuous media : accoustics, Euler,Navier-Stokes,...

Navier-Stokes equations :

∂tρ+∇x · (ρu) = 0,

ρ∂tu + ρu · ∇xu +∇xp = ν∆xu,

∂tE +∇x · (u (E + p)) = κ∆xθ,




Different scales of description

Mesoscopic description : collisional andmean-field models we are interested in the averagebehaviour of a particle, statistical/probabilist approach

We look at the dynamics of the particles density f (t, x , v)

Different modellings depending on the type of interactions :Vlasov, Vlasov-Poisson, Boltzmann...

Boltzmann equation :

∂t f + v · ∇x f = Q(f , f )




Coherence between different descriptions : Hilbert’s 6th

problem

Coherence between descriptions : we study the samesystem but on different scales

Microscopic Scale

Macroscopic Scale Mesoscopic Scale

N particles Hamiltonian systems

Continuous media Gas dynamics

(Newton’s laws)

(Euler, Navier-Stokes,. . . ) (Boltzmann equation)

Large deviations

N >> 1

Tcoll << Tobs

Hydrodynamical limit

Tcoll << Tobs

Thermodynamical limit

N >> 1

From L. Saint-Raymond Hydrodynamic limits of the Boltzmann equation




A mesoscopic point of viewThe collisional modelling of the Boltzmann equationA priori elementary properties of the Boltzmann equation

General presentation ofthe Boltzmann collisional

model





A probabilist approach to particle systems

Idea : Focus on the average behaviour of one particle in thesystem

Find an equation ruling the density function

f : [0,T ]× Td × Rd −→ R+

(t, x , v) 7−→ f (t, x , v)

f (t, x , v)dxdv is the probability of findinga particle inB(x , dx) with a velocity in B(v , dv) at time t

⇒ Minimal hypothesis : ∀t ∈ [0,T ], f (t, ·, ·) ∈ L1loc

(Ω, L1

v

(Rd))





The collisional process

1 Binary collisions : two particles that are sufficiently closeto each other are diviated (Boltzmann-Grad limit)

2 Localised collisions : trajectories are diverted veryrapidly, almost instantaneously

3 Elastic collisions :

v ′ + v ′∗ = v + v∗∣∣v ′∣∣2 +∣∣v ′∗∣∣2 = |v |2 + |v∗|2

4 Microreversible process : microscopic dynamics arereversible in time

5 Molecular chaos : particles involve independently





The Bolzmann equation

∀t > 0, ∀(x , v) ∈ Ω× Rd , ∂t f + v · ∇x f = Q(f , f )





The Boltzmann equation

The collisional operator :

Q(f , f ) =

∫Rd×Sd−1

|v − v∗|γ b (cos θ)[f ′f ′∗ − ff∗

]dv∗dσ

v ′ = v+v∗2 + |v−v∗|

2 σ

v ′∗ = v+v∗2 − |v−v∗|2 σ

, and cos θ = 〈 v−v∗|v−v∗| , σ〉.

γ belongs [0, 1] (hard and maxwellian potentials)b and b′ are bounded (strong Grad’s angular cut-off)





Physically observable quantities

local density :

ρ(t, x) =

∫Rd

f (t, x , v) dv

local velocity :

u(t, x) =1

ρ(t, x)

∫Rd

vf (t, x , v) dv

local energy :

E (t, x) =

∫Rd

|v |2

2f (t, x , v) dv = ρ(t, x)

|u|2

2+ d

ρ(t, x)θ(t, x)

2





Conservation laws

Preservation of the total mass

d

dt

∫Td

ρ(t, x) dx = 0,

Preservation of the total momentum

d

dt

∫Td

u(t, x) dx = 0,

Preservation of the total energy (thus the temperature)

d

dt

∫Td

E (t, x) dx = 0.





H-theorem and entropy dissipation

Entropy of the system :

S(f ) =

∫Td×Rd

f logf dxdv

Entropy dissipation :

D(f ) = −∫Rd

Q(f , f )logf dv

H-theorem :

d

dtS(f ) = −

∫Td

D(f ) dx 6 0.

⇒ The Boltzmann equation is time-irreversible !





Local and global equilibria

H-Theorem implies that global equilibria are the so-calledlocal Maxwellians :

M(ρ(t,x),u(t,x),θ(t,x))(v) =ρ

(2πθ)d/2e−|v−u|2

2θ .

Global equilibria :

∀(x , v) ∈ Td × Rd , v · ∇xM(ρ,u,θ) = 0.

In the torus it leads to independence on x so a unique globalequilibrium

µ(v) =1

(2π)d/2e−|v|2

2 .




From mesoscopic scale to macroscopic scaleDescription of the problemStudy of the linear part : hypocoercivityThe nonlinear Cauchy problem

The incompressibleNavier-Stokes limit of the

Boltzmann equation





Relation bewteen the two regimes

Dimensionless form of the Boltzmann equation :

∂t f + v .∇x f =1

εQ(f , f )

The Knudsen number : 1/ε stands for the averagenumber of collision undergone by one particle in a time unit

⇒ What are the physical quantities (ρ, u, θ) of f becomingwhen ε→ 0 ?

Rescalings and perturbative regime :

t → τ−1ε t

perturbation around global equilibrium fε = µ+ δεhε





Relation bewteen the two regimes

The perturbative equation :

∂thε +1

τεv .∇xhε =

1

ετεL(hε) +

δεετε

Q(hε, hε)

Formal convergence between models :(Bardos-Golse-Levermore)

If τε = δε and limε→0

δεε = 0 convergence towards incompressible

EulerIf τε = δε = ε convergence towards incompressibleNavier-StokesIf τε = ε and lim

ε→0

δεε = 0 towards incompressible Stokes

⇒ We are interested in the Navier-Stokes setting :

∂thε +1

εv .∇xhε =

1

ε2L(hε) +

1

εQ(hε, hε)





Description of the problem

H-Theorem : hε should tend to 0 as t → +∞A priori bounds on ‖hε‖t,x ,v that are uniform in ε would offerweak convergence of moments in v of hε

⇒ The goal is to establish a Cauchy theory together with anexponential trend to equilibrium independent of the Knudsen

number

Existing results :Cauchy problem for ε = 1 : in L2

v (Hsx )(µ−1/2

)(Ukai ’74), in

Hsx,v

(µ−1/2(1 + |v |)k

)(Guo ’03, Yu ’06), in

W p,kv W q,l

x (1 + |v |r ) (Gualdani-Mischler-Mouhot ’13)Exponential decay : in

(W α,1

v ∩W α,qv

)W β,p

x

((1 + |v |)k

)for

ε = 1 (hypocoercivite Mouhot-Neumann ’00,Gualdani-Mischler-Mouhot ’13) and uniformly in ε inHs

x,v

(µ−1/2

)(Guo ’05).





The linear Boltzmann operator

Properties in L2v

(µ−1/2

):

L is a closed self-adjoint operator L(f ) = −ν(v)f + K (f ),The collision frequency ν(v) ∼ (1 + |v |)γ ,K is a bounded kernel operator

Ker(L) =µ, vµ, |v |2 µ

;

The fluid part : othogonal projection onto the kernel of Lin L2

v

(µ−1/2

)πL(f )(t, x , v) =

[a(t, x) + b(t, x) · v + c(t, x)

∣∣v 2∣∣− 3

2

]µ(v);

The microscopic part :

π⊥L (f ) = f − πL(f )





Hypocoercivity in L2x ,v (µ−1/2)

Spectral gap :

〈L(f ), f 〉L2v(µ−1/2) > λL

∥∥∥π⊥L (f )∥∥∥2

L2v(νµ−1/2)

The linear equation : we define G = −v · ∇x + L andwe study

∂t f = G (f );

The transport is skew-symmetric therefore

d

dt‖f ‖2

L2x,v (µ−1/2) 6 −2

λLε2

∥∥∥π⊥L (f )∥∥∥2

L2x,v (νµ−1/2)





Hypocoercivity in L2x ,v (µ−1/2)

Recovering coercivity :Estimate πL(f ) thanks to π⊥L (f ) in the set of solutions to theequation,Idea of Guo (’00-’06) : estimating fluid equations

∆πL(f ) ∼ ∂2π⊥L (f ) + lower order terms

Idea of Mouhot-Neumann ’00 : hypocoercive norms

Problems and solutions :The transport and L do not have the same penalisation in ε,1st idea : a global hypocoercive norm

‖.‖2H1ε

= A ‖.‖2L2x,v

+α ‖∇x .‖2L2x,v

+bε2 ‖∇v .‖2L2x,v

+aε〈∇x .,∇v .〉L2x,v

2nd idea : an adapted hypocoercive norm

‖·‖2H1ε⊥

= A ‖·‖2L2x,v

+α ‖∇x ·‖2L2x,v

+b∥∥∥∇vπ

⊥L

∥∥∥2

L2x,v

+aε〈∇x ·,∇v ·〉L2x,v





Recent results on the linear part

Theorem

There exists 0 < εd 6 1 such that Gε = ε−2L− ε−1v · ∇x

generates a C 0-semigroup exponentially decreasing in

1 M.B. ’13 : Hsx ,v

(µ−1/2

)for any s in N∗ ;

2 M.B., S. Merino, C. Mouhot ’14 : W α,qv W β,p

x

(1 + |v |k

)for all p, q in [1,+∞], α 6 β and all

k > k∗q =3 +

√49− 48/q

2+ γ

(1− 1

q

).

⇒ The expontial decay rate is independent of ε !





General strategy

Control of the bilinear operator Q : requiresbounds in L∞ so we want to work in Hs with s > d/2

‖Q(f , g)‖Hsx,v (µ−1/2)

6 C[‖f ‖Hs

x,v (µ−1/2) ‖g‖Hsx,v (νµ−1/2) + ‖f ‖Hs

x,v (νµ−1/2) ‖g‖Hsx,v (µ−1/2)

]2 main problems and solutions :

Loss of weight ν ∼ 1 + |v |γ but the spectral gap compensateon the microscopic partNegative feedback of order O(1) of the linear operator becauseof the fluid part but we have a more subtle property for Q :

Q(f , g) ∈ (Ker(L))⊥

⇒ Require to include the nonlinear part inside the purely linearstudy !





Recent results on the nonlinear problem

Theorem

There exists 0 < εd 6 1 such that existence, uniqueness andexponential trend to equilibrium for small initial data :

1 M.B. ’13 : Hsx ,v

(µ−1/2

)for all s > s0 ;

2 M.B., S. Merino, C. Mouhot ’14 : W α,1v W β,p

x

(1 + |v |k

)for p = 1, 2, β > βp and α 6 β.

⇒ Uniformly in ε !




Quantitative resultsFourier analysis associated to the perturbation

Quantitative study of theconvergence towards

IncompressibleNavier-Stokes





Remarks on the previous theorem

ε-uniformity shows the weak-* convergence of (hε)ε inL∞t Hs

xL2v (µ−1/2) towards

h(t, x , v) =

[ρ(t, x) + v · u(t, x) +

1

2(|v |2 − d)θ(t, x)

]µ(v)

(ρ, u, θ) solution to Navier-Stokes satisfying div(u) = 0 andBoussinesq relation : ρ+ θ = 0

Still unresolved :

What is the speed of the latter convergence ?What about the associated initial data (ρ0, u0, θ0) ?





Speed of convergence and initial data

Theorem

Need of integrating in time over the torus :∥∥∥∥∫ +∞

0hε dt −

∫ +∞

0h dt

∥∥∥∥HsxL

2v (µ−1/2)

6 C√εlogε

Conditions on initial boundary layer : if(ρinε , u

inε , θ

inε ) satisfies incompressibility and Boussinesq

relations then

‖hε − h‖L2tH

sxL

2v (µ−1/2) 6 C

√εlogε

Improving the convergence : if hinε ∈ Hs+δ

x L2v then

supt∈R+

‖hε − h‖HsxL

2v (µ−1/2) 6 Cεminδ,1/2.





Link between convergence and Fourier spectrum

Under Duhamel form

hε = etGεhinε +

∫ t

0

(1

εe(t−s)Gε

)Q(hε(s), hε(s)) ds

Spatial Fourier transform and convergence : onthe torus the Fourier variable n is discrete

etGε = F−1x Uε(t, n, v)Fx

We recall Gε = 1ε2 L− 1

εv · ∇x therefore

Uε(t, n, v) = U1

( t

ε2, εn, v

)⇒ Need to know the behaviour of U1 at the origin





Spatial Fourier transform close to the origin :

Theorem (Ellis et Pinsky ’75)

There exists n0 > 0 and C∞ functions such that

U1(t, n, v) =2∑

j=−1

1|n|6n0etλj (|n|)Pj(n)(v) + UR(t, n, v)

and satisfying :

λj(|n|) = iαj |n| − βj |n|2 + γj(|n|) with α0,2 = 0 and

|γj(|n|)| 6 min

C |n|3 , βj2 |n|2

Pj(n) = P0j

(n|n|

)+ |n|P1j

(n|n|

)+ |n|2 P2j (n) and∑

P0,j = πL ∥∥∥UR

∥∥∥L2v

6 Ce−σt .Marc Briant From Boltzmann to Incompressible Navier-Stokes




Strategies for estimating convergence

We decompose

Uε1j(t, n, v) =e

iαj t|n|ε−βj t|n|2P0j

(n

|n|

)+ 1ε|n|6n0

eiαj t|n|ε−βj t|n|2

(e

tε2 γj (|εn|) − 1

)P0j

(n

|n|

)+ 1ε|n|6n0

eiαj t|n|ε−βj t|n|2+ t

ε2 γj (|εn|)ε |n| P1j

(|εn| , n

|n|

)+(1ε|n|6n0

− 1)

eiαj t|n|ε−βj t|n|2P0j

(n

|n|

)⇒ The first term is independent of ε for j = 0, 2 and gives rise to

the projection operator onto the space of null divergence+Boussinesq !





Future investigations

Develop a similar theory in spaces without derivative :unfortunately the standard methods of regularisation comeswith a factor 1/

√ε

What about the bounded domains with specular/diffusiveboundary conditions ? Existing proofs for ε = 1 are alreadyreally technical and not numerous (Guo, Kim-Yun,Esposito-Guo-Kim-Marra, M.B., M.B.-Guo).

THANKS FOR YOUR ATTENTION


FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES : HYDRODYNAMICAL LIMITS ... › 2018 › 03 ›...

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Transcript of FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES : HYDRODYNAMICAL LIMITS ... › 2018 › 03 ›...