IITBombay ∼ dolbeaulCeremade,UniversitéParis-DauphineOctober7,2019 JeanDolbeaulthttp://www.ceremade.dauphine.fr/ Hypocoercivity

Jean Dolbeault

http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Université Paris-Dauphine

October 7, 2019 IIT Bombay

Outline

From ϕ-entropies to H¹ hypocoercivity Bϕ-entropies and diffusions

Bϕ-hypocoercivity (H¹ framework)

An L² abstract result and mode-by-mode hypocoercivity BAbstract statement, toy model, global L² hypocoercivity result BDiffusion limit, application to the torus and a more numerical point of view

BDecay rates in the Euclidean space without confinement Diffusion and kinetic transport with very weak confinement

The Vlasov-Poisson-Fokker-Planck system: linearization and hypocoercivity

From ϕ-entropies to H ¹ hypocoercivity

BSome references of related works (Chafaï 2004),(Bolley, Gentil 2010) (Baudoin 2017)

(Monmarché),(Evans, 2017)

(Arnold, Erb, 2014),(Arnold, Stürzer), (Achleitner, Arnold, Stürzer, 2016),(Achleitner, Arnold, Carlen, 2017),(Arnold, Einav, Wöhrer, 2017)

BIn collaboration with X. Li

Definition of the ϕ-entropies

E[w] :=

Z

R^d

ϕ(w)dγ

ϕis a nonnegative convex continuous function onR⁺ such that ϕ(1) = 0 and 1/ϕ⁰⁰is concave on (0,+∞):

ϕ⁰⁰≥0, ϕ≥ϕ(1) = 0 and (1/ϕ⁰⁰)⁰⁰≤0 Classical examples

ϕ_p(w) :=_p−1¹ w^p−1−p(w−1)

p∈(1,2]

ϕ₁(w) :=w logw−(w−1) The invariant measure

dγ=e^−ψdx

whereψis apotentialsuch thate^−ψ is in L¹(R^d,dx) dγ is a probability measure

Diffusions

Ornstein-Uhlenbeck equationor backward Kolmogorov equation

∂w

∂t =Lw:= ∆w− ∇ψ· ∇w

− Z

R^d

(Lw1)w2dγ= Z

R^d

∇w1· ∇w2dγ ∀w1,w2∈H¹(R^d,dγ) 1 =

Z

R^d

w₀dγ= Z

R^d

w(t,·)dγ and lim_t→+∞w(t,·) = 1 d

dtE[w] =− Z

R^d

ϕ⁰⁰(w)|∇xw|²dγ=:−I[w] (Fisher information) If for some Λ>0: entropy – entropy production inequality

I[w]≥ΛE[w] ∀w∈H¹(R^d,dγ) E[w(t,·)]≤E[w0]e^−Λt ∀t≥0 Fokker-Planck equation: u=wγ converges tou?=γ

∂u

∂t = ∆u+∇x·(u∇xψ)

Generalized Csiszár-Kullback-Pinsker inequality

(Pinsker),(Csiszár 1967),(Kullback 1967),(Cáceres, Carrillo, JD, 2002)

Proposition

Let p∈[1,2], w∈L¹∩L^p(R^d,dγ)be a nonnegative function, and assume thatϕ∈C²(0,+∞)is a nonnegative strictly convex function such thatϕ(1) =ϕ⁰(1) = 0. If A:= inf_s∈(0,∞)s^2−pϕ⁰⁰(s)>0, then

E[w]≥2^−2pAminn

1,kwk^p−2_Lp(R^d,dγ)

okw−1k²_Lp(R^d,dγ)

Convexity, tensorization and sub-additivity

Z

R^di

ϕ⁰⁰(w)|∇w|²dγ_i =:Iγi[w]≥Λ_iEγi[w] ∀w∈H¹(R^di,dγ_i)

Theorem

If dγ1 and dγ2 are two probability measures on R^d1×R^d2, then Iγ₁⊗γ2[w] =

Z

R^d1×R^d2

ϕ⁰⁰(w)|∇w|²dγ1dγ2

≥min{Λ₁,Λ₂}Eγ1⊗γ2[w] ∀w∈H¹(R^d1×R^d2,dγ)

Iγ1⊗γ2[w] = Z

R^d2

Iγ1[w]dγ2+ Z

R^d1

Iγ2[w]dγ1

Eγ₁⊗γ2[w]≤ Z

R^d2

Eγ₁[w]dγ2+ Z

R^d1

Eγ₂[w]dγ1 ∀w∈L¹(dγ1⊗γ2)

Perturbation (Holley-Stroock type) results

Withw:=R

R^dw dγ, assume that Λ

Z

R^d

ϕ(w)dγ−ϕ(w)

≤ Z

R^d

ϕ⁰⁰(w)|∇w|²dγ ∀w∈H¹(dγ) and, for some constantsa, b∈R,

e^−bdγ≤dµ≤e^−adγ Lemma

Ifϕis a C² function such thatϕ⁰⁰>0andwe:=R

R^dw dµ /R

R^ddµ, then

e^a−bΛ Z

R^d

ϕ(w)−ϕ(w)e −ϕ⁰(w)(we −w)e dµ≤

Z

R^d

ϕ⁰⁰(w)|∇w|²dµ

Entropy – entropy production inequalities, linear flows

On a smooth convex bounded domain Ω, consider

∂w

∂t =Lw:= ∆w− ∇ψ· ∇w, ∇w·ν = 0 on ∂Ω d

dt Z

Ω

w^p−1

p−1 dγ=−4 p

Z

Ω

|∇z|²dγ and z =w^p/2 d

dt Z

Ω

|∇z|²dγ≤ −2 Λ(p) Z

Ω

|∇z|²dγ where Λ(p)>0 is the best constant in the inequality

2 p(p−1)

Z

Ω

|∇X|²dγ+ Z

Ω

Hessψ:X⊗X dγ≥Λ(p) Z

Ω

|X|²dγ Proposition

Z

Ω

w^p−1

p−1 dγ≤ 4 pΛ

Z

Ω

|∇w^p/2|²dγ for any w s.t.

Z

Ω

w dγ= 1 (Bakry, Emery 1985)

An interpolation inequality

Corollary

Assume that q∈[1,2). WithΛ = Λ(2/q), we have kfk²_L2(R^d,dγ)− kfk²_Lq(R^d,dγ)

2−q ≤ 1

Λ Z

R^d

|∇f|²dγ ∀f ∈H¹(R^d,dγ)

Improved entropy – entropy production inequalities

In the special caseψ(x) =|x|²/2, withz =w^p/2, we obtain that 1

2 d dt

Z

R^d

|∇z|²dγ+ Z

R^d

|∇z|²dγ≤ −2 pκ_p

Z

R^d

|∇z|⁴ z² dγ withκ_p= (p−1) (2−p)/p

Cauchy-Schwarz: R

R^d|∇z|²dγ2

≤R

R^d

|∇z|⁴ z² dγR

R^dz²dγ d

dtI[w] + 2I[w]≤ −κp I[w]² 1 + (p−1)E[w]

Proposition

Assume that q∈(1,2) and dγ= (2π)^−d/2e^−|x|2/2dx. There exists a strictly convex function F such that F(0) = 0 and F⁰(0) = 1 and

F

kfk²_L2(R^d,dγ)−1

≤ k∇fk²_L2(R^d,dγ) if kfk_Lq(R^d,dγ)= 1

ϕ-hypocoercivity (H ¹ framework)

Badapt the strategy ofϕ-entropies to kinetic equations

BVillani’s strategy: derive H¹estimates (using a twisted Fisher information) and then use standard interpolation inequalities to establish entropy decay rates

The twisted Fisher informationis notthe derivative of theϕ-entropy

Thekinetic Fokker-Planck equation, orVlasov-Fokker-Planck equation:

∂f

∂t +v· ∇_xf− ∇_xψ· ∇_vf = ∆_vf+∇_v·(v f) (1) withψ(x) =|x|²/2 andkfk_L1(R^d×R^d)= 1 has a unique nonnegative stationary solution

f?(x,v) = (2π)^−de⁻

1

2(|x|²+|v|²)

and the functiong=f/f_? solves thekinetic Ornstein-Uhlenbeck equation

∂g

∂t +Tg=Lg

with transport operatorTand Ornstein-Uhlenbeck operatorLgiven by

Tg:=v· ∇_xg−x· ∇_vg and Lg:= ∆_vg−v· ∇_vg

Sharp rates for the kinetic Fokker-Planck equation

Letψ(x) =|x|²/2,dµ:=f?dx dv,E[g] :=

Z Z

R^d×R^d

ϕp(g)dµ Proposition

Let p∈[1,2]and consider a nonnegative solution g∈L¹(R^d×R^d) of the kinetic Fokker-Planck equation. There is a constantC>0such that

E[g(t,·,·)]≤Ce^−t ∀t ≥0 and the rate e^−t is sharp as t→+∞

(Villani),(Arnold, Erb): a twistedFisher informationfunctional Jλ[h] = (1−λ)

Z

R^d

|∇vh|²dµ+(1−λ) Z

R^d

|∇xh|²dµ+λ Z

R^d

|∇xh+∇vh|²dµ (Arnold, Erb)relies onλ= 1/2 and _dt^dJ1/2[h(t,·)]≤ −J1/2[h(t,·)]

Improved rates (in the large entropy regime)

Rewrite the decay of theFisher informationfunctional as

−¹₂ d dt

Z

R^d

X^⊥·M0X dµ= Z

R^d

X^⊥·M1X dµ+ Z

R^d

Y^⊥·M2Y dµ whereX = (∇vh,∇xh), Y = (Hvv,Hxv,Mvv,Mxv)

M₀=

1 λ λ ν

⊗Id_Rd, M₁=

1−λ ^1+λ−ν₂

1+λ−ν

2 λ

⊗ Id_Rd

M2=









1 λ −^κ₂ −^{κ λ}₂ λ ν −^{κ λ}₂ −^{κ ν}₂

−^κ₂ −^{κ λ}₂ 2κ 2κ λ

−^{κ λ}₂ −^{κ ν}₂ 2κ λ 2κν









⊗Id_Rd×R^d

With constant coefficients λ?(λ, ν) = max

min

X

X^⊥·M1X

X^⊥·M₀X : (λ, ν)∈R²s.t.M2≥0

For (λ, ν) = (1/2), λ?= 1/2 and the eigenvalues ofM₂(¹₂,1) are given as a function ofκ= 8 (2−p)/p∈[0,8] are all nonnegative

2 4 6 8

0.5 1.0 1.5

λ1(κ)

λ2(κ) λ3(κ)

λ4(κ)

κ

We know that

Y^⊥·M₂Y ≥λ₁(p, λ)|Y|²

for someλ1(p, λ)>0 and|Y|²≥ kMvvk²so that, by Cauchy-Schwarz, Z

R^d

|∇_vh|²dµ ²

≤ Z

R^d

h²dµ Z

R^d

kM_vvk²dµ≤c₀ Z

R^d

kM_vvk²dµ Theorem

Let p∈(1,2)and h be a solution of the kinetic Ornstein-Uhlenbeck equation. Then there exists a functionλ:R⁺→[1/2,1)such that λ(0) = lim_t→+∞λ(t) = 1/2 and a function ρ >1/2 s.t.

d

dtJλ(t)[h(t,·)]≤ −2ρ(t)Jλ(t)[h(t,·)]

As a consequence, for any t≥0 we have the global estimate Jλ(t)[h(t,·)]≤J1/2[h0] exp

−2 Z t

0

ρ(s)ds

Let us definea:=e^tR

R^d|∇vh|²dµ,b:=e^tR

R^d∇vh· ∇xh dµ, c:=e^tR

R^d|∇xh|²dµandj:=a+b+c da

dt ≤a−2 (j−c), dc

dt ≤2 (j−a)−c and dj dt ≤0 with the constraintsa≥0,c≥0 andb²≤a c

0.0 0.5 1.0 1.5 2.0

J. Dolbeault Hypocoercivity

An abstract hypocoercivity result and mode-by-mode hypocoercivity

BAbstract statement, toy model and a global L²hypocoercivity result

BMode-by-mode hypocoercivity

BApplication to the torus and numerics BDecay rates in the whole space

Collaboration with E. Bouin, S. Mischler, C. Mouhot, C. Schmeiser

An abstract evolution equation

Let us consider the equation dF

dt +TF =LF (2)

In the framework of kinetic equations,TandLare respectively the transport and the collision operators

We assume thatTandL are respectively anti-Hermitian and Hermitian operators defined on the complex Hilbert space (H,h·,·i)

A:= 1 + (TΠ)^∗TΠ⁻¹ (TΠ)^∗

∗denotes the adjoint with respect toh·,·i

Π is the orthogonal projection onto the null space ofL

The assumptions

λm, λM, andCM are positive constants such that, for anyF ∈H Bmicroscopic coercivity:

− hLF,Fi ≥λ_mk(1−Π)Fk² (H1) Bmacroscopic coercivity:

kTΠFk²≥λ_MkΠFk² (H2) Bparabolic macroscopic dynamics:

ΠTΠF = 0 (H3)

Bbounded auxiliary operators:

kAT(1−Π)Fk+kALFk ≤C_Mk(1−Π)Fk (H4) The estimate

1 2

d

dtkFk²=hLF,Fi ≤ −λ_mk(1−Π)Fk² is not enough to conclude thatkF(t,·)k² decays exponentially

Equivalence and entropy decay

For someδ >0 to be determined later, the L² entropy / Lyapunov functional is defined by

H[F] := ¹₂kFk²+δRehAF,Fi

as in(J.D.-Mouhot-Schmeiser) so thathATΠF,Fi ∼ kΠFk² and

− d

dtH[F] = :D[F]

=− hLF,Fi+δhATΠF,Fi

−δRehTAF,Fi+δRehAT(1−Π)F,Fi −δRehALF,Fi

Bentropy decay rate: for anyδ >0 small enough andλ=λ(δ) λH[F]≤D[F]

Bnorm equivalenceofH[F] and kFk² 2− δ

4 kFk²≤H[F]≤2 +δ 4 kFk²

Exponential decay of the entropy

λ= _{3 (1+λ}^λM

M)minn

1, λm,_(1+λ^λmλM

M)C_M²

o

,δ= ¹₂ min

1, λm,_(1+λ^λmλM

M)C_M²

h₁(δ, λ) := (δC_M)²−4

λ_m−δ−2 +δ

4 λ δ λ_M

1 +λM

−2 +δ 4 λ

Theorem

LetL andTbe closed linear operators (respectively Hermitian and anti-Hermitian) onH. Under (H1)–(H4), for any t ≥0

H[F(t,·)]≤H[F0]e^−λ?t whereλ? is characterized by

λ?:= sup

λ >0 : ∃δ >0s.t. h1(δ, λ) = 0, λm−δ−¹₄(2 +δ)λ >0

Sketch of the proof

SinceATΠ = 1 + (TΠ)^∗TΠ⁻¹

(TΠ)^∗TΠ, from (H1) and (H2)

− hLF,Fi+δhATΠF,Fi ≥λmk(1−Π)Fk²+ δ λM

1 +λ_M kΠFk² By (H4), we know that

|RehAT(1−Π)F,Fi+ RehALF,Fi| ≤CMkΠFk k(1−Π)Fk The equationG=AF is equivalent to (TΠ)^∗F=G+ (TΠ)^∗TΠG hTAF,Fi=hG,(TΠ)^∗Fi=kGk²+kTΠGk²=kAFk²+kTAFk² hG,(TΠ)^∗Fi ≤ kTAFk k(1−Π)Fk ≤ 1

2µkTAFk²+µ

2 k(1−Π)Fk² kAFk ≤ 1

2k(1−Π)Fk, kTAFk ≤ k(1−Π)Fk, |hTAF,Fi| ≤ k(1−Π)Fk² WithX :=k(1−Π)Fkand Y :=kΠFk

D[F]−λH[F]≥(λ_m−δ)X²+ δ λ_M 1 +λM

Y²−δC_MX Y−2 +δ

4 λ(X²+Y²)

Hypocoercivity

Corollary

For anyδ∈(0,2), ifλ(δ)is the largest positive root of h1(δ, λ) = 0for whichλm−δ−¹₄(2 +δ)λ >0, then for any solution F of (2)

kF(t)k²≤ 2 +δ

2−δe^−λ(δ)tkF(0)k² ∀t≥0 From the norm equivalence ofH[F] andkFk²

2− δ

4 kFk²≤H[F]≤2 +δ 4 kFk² We use ²⁻₄^δkF0k²≤H[F0] so thatλ?≥sup_δ∈(0,2)λ(δ)

Formal macroscopic (diffusion) limit

Scaled evolution equation εdF

dt +TF =1 εLF

on the Hilbert spaceH. F_ε=F₀+εF₁+ε²F₂+O(ε³) asε→0₊ ε⁻¹: LF0= 0,

ε⁰: TF0=LF1, ε¹: ^dF_dt⁰ +TF1=LF2

The first equation reads asF0= ΠF0

The second equation is simply solved byF1=−(TΠ)F0

After projection, the third equation is

d

dt(ΠF₀)− ΠT(TΠ)F₀= ΠLF₂= 0

∂_tu+ (TΠ)^∗(TΠ)u= 0 is such that _dt^dkuk²=−2k(TΠ)uk²≤ −2λMkuk²

A toy problem

du

dt = (L−T)u, L=

0 0 0 −1

, T=

0 −k

k 0

, k²≥Λ>0 Non-monotone decay, a well known picture:

see for instance(Filbet, Mouhot, Pareschi, 2006) H-theorem: _dt^d|u|²=−2u₂²

macroscopic limit: ^du_dt¹ =−k²u1

generalized entropy: H(u) =|u|²− _1+k^δk2u1u2

dH

dt = −

2− δk² 1 +k²

u₂²− δk²

1 +k²u²₁+ δk 1 +k²u₁u₂

≤ −(2−δ)u₂²− δΛ

1 + Λu²₁+δ 2u₁u₂

Plots for the toy problem

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1 u1²

1 2 3 4 5 6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 u1²+u2²

1 2 3 4 5 6

0.2 0.4 0.6 0.8 u11²

u1² u2²

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1 H

Mode-by-mode hypocoercivity

BFokker-Planck equation and scattering collision operators BA mode-by-mode hypocoercivity result

BEnlargement of the space by factorization

BApplication to the torus and some numerical results

(Bouin, J.D., Mischler, Mouhot, Schmeiser)

Fokker-Planck equation with general equilibria

We consider the Cauchy problem

∂_tf +v· ∇xf =Lf, f(0,x,v) =f0(x,v) (3) for a distribution functionf(t,x,v), withpositionvariablex∈R^d or x∈T^d the flat d-dimensional torus

Fokker-Planckcollision operator with a general equilibriumM Lf =∇v·h

M∇v M⁻¹fi

Notation and assumptions: anadmissible local equilibrium M is positive, radially symmetric and

Z

R^d

M(v)dv= 1, dγ=γ(v)dv:= dv M(v) γis anexponential weightif

lim

|v|→∞

|v|^k

γ(v)= lim

|v|→∞M(v)|v|^k= 0 ∀k∈(d,∞)

Definitions

Θ = 1 d

Z

R^d

|v|²M(v)dv= Z

R^d

(v·e)²M(v)dv for an arbitrarye∈S^d−1

Z

R^d

v⊗v M(v)dv= Θ Id Then

θ= 1

d k∇vMk²_L2(dγ)= 4 d

Z

R^d

|∇v

√ M

2dv<∞ IfM(v) = ^e−

1 2|v|2

(2π)^d/2, then Θ = 1 andθ= 1 σ:= 1

2 pθ/Θ

Microscopic coercivity property(Poincaré inequality): for all u=M⁻¹F∈H¹(M dv)

Z

R^d

|∇u|²M dv≥λm

Z

R^d

u−

Z

R^d

u M dv ²

M dv

Scattering collision operators

Scatteringcollision operator Lf =

Z

R^d

σ(·,v⁰) f(v⁰)M(·)−f(·)M(v⁰) dv⁰

Main assumption on thescattering rateσ: for some positive, finiteσ 1≤σ(v,v⁰)≤σ ∀v,v⁰∈R^d

Example: linear BGK operator

Lf =Mρf−f, ρf(t,x) = Z

R^d

f(t,x,v)dv Local mass conservation

Z

R^d

Lf dv = 0 and we have

Z

R^d

|Lf|²dγ≤4σ² Z

R^d

|Mρf−f|²dγ

The symmetry condition Z

R^d

σ(v,v⁰)−σ(v⁰,v)

M(v⁰)dv⁰ = 0 ∀v∈R^d implies thelocal mass conservationR

R^dLf dv= 0

Micro-reversibility,i.e., the symmetry ofσ, is not required The null space ofLis spanned by the local equilibriumM Lonly acts on the velocity variable

Microscopic coercivity property: for someλm>0 1

2 Z Z

R^d×R^d

σ(v,v⁰)M(v)M(v⁰) (u(v)−u(v⁰))²dv⁰dv

≥λm

Z

R^d

(u−ρu M)²M dv holds according to Proposition 2.2 of(Degond, Goudon, Poupaud, 2000)for allu=M⁻¹F ∈L²(M dv). Ifσ≡1, thenλ_m= 1

Fourier modes

In order to perform amode-by-mode hypocoercivityanalysis, we introduce the Fourier representation with respect tox,

f(t,x,v) = Z

R^d

ˆf(t, ξ,v)e^{−i x·ξ}dµ(ξ)

dµ(ξ) = (2π)^−ddξ anddξ is the Lesbesgue measure ifx∈R^d dµ(ξ) = (2π)^−d P

z∈Z^dδ(ξ−z) is discrete forx∈T^d

Parseval’s identity ifξ∈Z^d and Plancherel’s formula ifx ∈R^d read kf(t,·,v)k_L2(dx)=

ˆf(t,·,v)

_L2(dµ(ξ)) The Cauchy problem is now decoupled in theξ-direction

∂tˆf +Tˆf =Lˆf, ˆf(0, ξ,v) = ˆf0(ξ,v) Tˆf =i(v·ξ) ˆf

For any fixedξ∈R^d, let us apply the abstract result with H= L²(dγ), kFk²=

Z

R^d

|F|²dγ , ΠF =M Z

R^d

F dv =MρF

andTˆf =i(v·ξ) ˆf,TΠF=i(v·ξ)ρFM, kTΠFk²=|ρ_F|²

Z

R^d

|v·ξ|²M(v)dv = Θ|ξ|²|ρ_F|²= Θ|ξ|²kΠFk² (H2)Macroscopic coercivitykTΠFk²≥λ_MkΠFk² : λ_M= Θ|ξ|² (H3)R

R^dv M(v)dv= 0 The operatorAis given by

AF =−iξ·R

R^dv⁰F(v⁰)dv⁰ 1 + Θ|ξ|² M

A mode-by-mode hypocoercivity result

kAFk=kA(1−Π)Fk ≤ 1 1 + Θ|ξ|²

Z

R^d

|(1−Π)F|

√

M |v·ξ|√ M dv

≤ 1

1 + Θ|ξ|²k(1−Π)Fk Z

R^d

(v·ξ)²M dv 1/2

=

√ Θ|ξ|

1 + Θ|ξ|²k(1−Π)Fk Scattering operatorkLFk²≤4σ²k(1−Π)Fk²

Fokker-Planck (FP) operator kALFk ≤ 2

1 + Θ|ξ|² Z

R^d

|(1−Π)F|

√

M |ξ·∇_v√

M|dv≤

√θ|ξ|

1 + Θ|ξ|²k(1−Π)Fk In both cases withκ=√

θ(FP) orκ= 2σ√

Θ we obtain kALFk ≤ κ|ξ|

1 + Θ|ξ|²k(1−Π)Fk

TAF(v) =−(v·ξ)M 1 + Θ|ξ|²

Z

R^d

(v⁰·ξ) (1−Π)F(v⁰)dv⁰ is estimated by

kTAFk ≤ Θ|ξ|²

1 + Θ|ξ|²k(1−Π)Fk (H4) holds withCM= ^κ_1+Θ^|ξ|+Θ_|ξ|^|ξ|2²

Two elementary estimates Θ|ξ|²

1 + Θ|ξ|² ≥ Θ max{1,Θ}

|ξ|² 1 +|ξ|² λM

(1 +λ_M)C_M² = Θ (1 + Θ|ξ|²)

(κ+ Θ|ξ|)² ≥ Θ κ²+ Θ

Mode-by-mode hypocoercivity with exponential weights

Theorem

Let us consider an admissible M and a collision operatorLsatisfying Assumption(H), and takeξ∈R^d. Ifˆf is a solution such that ˆf0(ξ,·)∈L²(dγ), then for any t≥0, we have

ˆf(t, ξ,·)

2 L²(dγ)

≤3e^−µξt

ˆf₀(ξ,·)

2 L²(dγ)

where

µξ := Λ|ξ|²

1 +|ξ|² and Λ = Θ

3 max{1,Θ} min

1, λmΘ κ²+ Θ

withκ= 2σ√

Θfor scattering operators andκ=√

θfor (FP) operators

Exponential convergence to equilibrium in T

The unique global equilibrium in the casex∈T^d is given by f∞(x,v) =ρ∞M(v) with ρ∞= 1

|T^d| Z Z

T^d×R^d

f0dx dv

Theorem

Assume thatγ has an exponential growth. We consider an admissible M , a collision operatorLsatisfying Assumption (H). There exists a positive constant C such that the solution f of (3) onT^d×R^d with initial datum f0∈L²(dx dγ)satisfies

kf(t,·,·)−f_∞k_L2(dx dγ)≤C kf0−f_∞k_L2(dx dγ)e^−14Λt ∀t≥0

Enlargement of the space by factorization

A simple case (factorization of order 1) of thefactorization method of(Gualdani, Mischler, Mouhot)

Theorem

LetB1,B2 be Banach spaces and letB2 be continuously imbedded in B1, i.e.,k · k₁≤c₁k · k₂. Let BandA+B be the generators of the strongly continuous semigroups e^Bt and e^(A+B)t onB1. If for all t≥0,

e^(A+B)t

2→2≤c2e^−λ2t, e^Bt

_1→1≤c3e^−λ1t, kAk_1→2≤c4

wherek · ki→j denotes the operator norm for linear mappings fromBi

toBj. Then there exists a positive constant C =C(c1,c2,c3,c4)such that, for all t≥0,

e^(A+B)t _1→1≤

( C(1 +|λ₁−λ₂|⁻¹)e^{−min{λ1,λ2}t} forλ₁6=λ₂ C(1 +t)e^−λ1t forλ₁=λ₂

Integrating the identity _ds^d e^(A+B)se^B(t−s)

=e^(A+B)sAe^B(t−s) with respect tos∈[0,t] gives

e^(A+B)t =e^Bt+ Z t

0

e^(A+B)sAe^B(t−s)ds The proof is completed by the straightforward computation

e^(A+B)t

1→1≤c₃e^−λ1t+c₁ Z t

0

e^(A+B)sAe^B(t−s) 1→2ds

≤c₃e^−λ1t+c₁c₂c₃c₄e^−λ1t Z t

0

e^(λ1−λ2)sds

Weights with polynomial growth

Let us consider the measure

dγ_k:=γ_k(v)dv where γ_k(v) =πd/2 Γ((k−d)/2)

Γ(k/2) 1 +|v|^2k/2 for an arbitraryk∈(d,+∞)

We chooseB1= L²(dγ_k) andB2= L²(dγ) Theorem

LetΛ = _{3 max{1,Θ}}^Θ minn 1,_κ^λ2^m+Θ^Θ

o

and k∈(d,∞]. For anyξ∈R^d if ˆf is a solution with initial datumˆf0(ξ,·)∈L²(dγk), then there exists a constant C =C(k,d, σ)such that

ˆf(t, ξ,·)

2 L²(dγk)

≤C e^−µξt

ˆf₀(ξ,·)

2 L²(dγk)

∀t≥0

Fokker-Planck: AF=Nχ_RF andBF =−i(v·ξ)F+LF−AF N andRare two positive constants,χis a smooth cut-off function andχ_R:=χ(·/R)

For anyRandN large enough, according to Lemma 3.8 of(Mischler, Mouhot, 2016)

Z

R^d

(L−A)(F)F dγ_k ≤ −λ₁ Z

R^d

F²dγ_k for someλ1>0 ifk>d, andλ2=µξ/2≤1/4

Scattering operator:

AF(v) = M(v) Z

R^d

σ(v,v⁰)F(v⁰)dv⁰ BF(v) = −

i(v·ξ) + Z

R^d

σ(v,v⁰)M(v⁰)dv⁰

F(v) Boundedness: kAFk_L2(dγ)≤σ R

R^dγ⁻¹_k dv^1/2

kFk_L2(dγk)

λ1= 1 andλ2=µξ/2≤1/4

Exponential convergence to equilibrium in T

The unique global equilibrium in the casex∈T^d is given by f_∞(x,v) =ρ_∞M(v) with ρ_∞= 1

|T^d| Z Z

T^d×R^d

f₀dx dv

Theorem

Assume that k∈(d,∞] andγ has an exponential growth if k=∞.

We consider an admissible M , a collision operatorLsatisfying Assumption(H), andΛgiven by (11)

There exists a positive constant Ck such that the solution f of (3) on T^d×R^d with initial datum f0∈L²(dx dγk) satisfies

kf(t,·,·)−f∞k_L2(dx dγ_k)≤Ck kf0−f∞k_L2(dx dγ_k)e^−14Λt ∀t ≥0

If we represent the flat torusT^d by the box [0,2π)^d with periodic boundary conditions, the Fourier variable satisfiesξ∈Z^d. Forξ= 0, the microscopic coercivity implies

ˆf(t,0,·)−ˆf_∞(0,·)

_L2(dγ)≤

ˆf₀(0,·)−ˆf_∞(0,·)

_L2(dγ)e^−t Otherwiseµξ ≥Λ/2 for anyξ6= 0

Parseval’s identity applies, with measuredγ(v) andC∞=√ 3 The result with weightγk follows from the factorization result for someCk>0

Computation of the constants

BA more numerical point of view

Two simple examples: Ldenotes either theFokker-Planck operator L₁f := ∆_vf+∇_v·(v f)

or thelinear BGK operator

L2f := Πf −f

Πf =ρfM is the projection operator on the normalized Gaussian function

M(v) = e^−12|v|2 (2π)^d/2 andρf :=R

R^df dv is the spatial density

Where do we have space for improvements ?

WithX :=k(1−Π)Fkand Y :=kΠFk, we wrote D[F]−λH[F]

≥(λm− δ)X²+ δ λM

1 +λM

Y²−δCMX Y −λ

2 X²+Y²+δX Y

≥(λm−δ)X²+ δ λM

1 +λM

Y²− δCMX Y −2 +δ

4 λ(X²+Y²) We can directly study the positivity condition for the quadratic form

(λm− δ)X²+ δ λM

1 +λM

Y²−δCMX Y −λ

2 X²+Y²+δX Y λm = 1,λM =|ξ|² andCM =|ξ|(1 +|ξ|)/(1 +|ξ|²)

2 4 6 8 10 0.1

0.2 0.3 0.4

Withλm= 1, λM =|ξ|² andCM =|ξ|(1 +|ξ|)/(1 +|ξ|²), we optimizeλunder the condition that the quadratic form

(λm− δ)X²+ δ λM

1 +λM

Y²−δCMX Y −λ

2 X²+Y²+δX Y is positive, thus getting aλ(ξ)

By taking alsoδ=δ(ξ) whereξis seen as a parameter, we get a better estimate ofλ(ξ)

2 4 6 8 10 0.1

0.2 0.3 0.4 0.5

By takingδ=δ(ξ), for each value ofξ we build a different Lyapunov function, namely

H_ξ[F] := ¹₂kFk²+δ(ξ) RehAF,Fi where the operatorAis given by

AF =−iξ·R

R^dv⁰F(v⁰)dv⁰ 1 +|ξ|² M We can consider

AεF =−iξ·R

R^dv⁰F(v⁰)dv⁰ ε+|ξ|² M and look for the optimal value ofε...

2 4 6 8 10 0.1

0.2 0.3 0.4 0.5

The dependence ofλin εis monotone, and the limit asε→0+ gives the optimal estimate ofλ. The operator

A₀F =−iξ·R

R^dv⁰F(v⁰)dv⁰

|ξ|² M

is not bounded anymore, but estimates still make sense and lim_ξ→0δ(ξ) = 0 (see below)

2 4 6 8 10

0.05 0.10 0.15 0.20 0.25 0.30 0.35

2 4 6 8 10 0.1

0.2 0.3 0.4

Theorem (Hypocoercivity onT^d with exponential weight) Assume thatL=L1 or L=L2. If f is a solution, then

kf(t,·,·)−f_∞k²_L2(dx dγ)≤C?kf0k²_L2(dx dγ)e^−λ?t ∀t≥0 with f_∞(x,v) =M(v)RR

T^d×R^df0(x,v)dx dv C?≈1.75863 andλ_?= ₁₃²(5−2√

3)≈0.236292.

(work in progress)

Some comments on recent works

A more algebraic approach based on the spectral analysis of symmetric and non-symmetric operators

On BGK models

(Achleitner, Arnold, Carlen) On Fokker-Planck models (Arnold, Erb)

(Arnold, Stürzer) (Arnold, Einav, Wöhrer)

Decay rates in the whole space

Algebraic decay rates in R

On the whole Euclidean space, we can define the entropy H[f] := ¹₂kfk²_L2(dx dγk)+δhAf,fidx dγk

Replacing themacroscopic coercivitycondition byNash’s inequality kuk²_L2(dx)≤CNashkuk

4 d+2

L¹(dx)k∇uk

2d d+2

L²(dx)

proves that

H[f]≤C

H[f0] +kf0k²_L1(dx dv)

(1 +t)^−d2 Theorem

Assume thatγk has an exponential growth (k =∞) or a polynomial growth of order k>d

There exists a constant C >0 such that, for any t≥0 kf(t,·,·)k²_L2(dx dγk)≤C

kf0k²_L2(dx dγk)+kf0k²_L2(dγk; L¹(dx))

(1 +t)^−d2

A direct proof... Recall thatµ_ξ= _1+|ξ|^Λ|ξ|2₂ By the Plancherel formula

kf(t,·,·)k²_L2(dx dγ_k)≤C Z

R^d

Z

R^d

e^−µξt|ˆf0|²dξ

dγk

if|ξ|<1, thenµξ ≥^Λ₂ |ξ|² Z

|ξ|≤1

e^−µξt|ˆf0|²dξ≤C kf0(·,v)k²_L1(dx)

Z

R^d

e^−Λ2|ξ|2tdξ

≤C kf0(·,v)k²_L1(dx)t^−d2 if|ξ| ≥1, thenµ_ξ ≥Λ/2 when |ξ| ≥1

Z

|ξ|>1

e^−µξt|ˆf₀|²dξ≤C e^−Λ2t kf₀(·,v)k²_L2(dx)

Improved decay rate for zero average solutions

Theorem

Assume that f0∈L¹_loc(R^d×R^d)withRR

R^d×R^df0(x,v)dx dv= 0 and C0:=kf0k²_L2(dγ_k+2; L¹(dx))+kf0k²_L2(dγ_k; L¹(|x|dx))+kf0k²_L2(dx dγ_k)<∞ Then there exists a constant ck>0 such that

kf(t,·,·)k²_L2(dx dγ_k)≤ckC0(1 +t)⁻(^1+d2)