Abstract method and motivation

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Jean Dolbeault

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

November 6, 2019 FLACAM 2019 – PDE session French Latin - American Conference on New Trends in Applied Mathematics

5 - 8 November 2019, Santiago, Chile

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Outline

Abstract method and motivation BAbstract statement in a Hilbert space BDiffusion limit, toy model

The compact case BStrong confinement

BMode-by-mode decomposition BApplication to the torus BFurther results

The non-compact case

BWithout confinement: Nash inequality

BWith very weak confinement: Caffarelli-Kohn-Nirenberg inequality

BWith sub-exponential equilibria: weighted Poincaré inequality The Vlasov-Poisson-Fokker-Planck system

BLinearized system and hypocoercivity

BResults in the diffusion limit and in the non-linear case

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Vlasov-Poisson-Fokker-Planck

Abstract method and motivation

BAbstract statement BDiffusion limit BA toy model

Collaboration with C. Mouhot and C. Schmeiser

J. Dolbeault Hypocoercivity and functional inequalities

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An abstract evolution equation

Let us consider the equation dF

dt +TF =LF

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

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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 ≤CMk(1−Π)Fk (H4) The estimate

1 2

d

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

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Equivalence and entropy decay

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

H[F] := ¹₂kFk²+δRehAF, Fi 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] andkFk² 2−δ

4 kFk²≤H[F]≤2 +δ 4 kFk²

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Exponential decay of the entropy

λ= _{3 (1+λ}^λ^M

M)minn

1, λm,_(1+λ^λ^m^λ^M

M)C_M²

o

,δ=¹₂ min

1, λm,_(1+λ^λ^m^λ^M

M)C_M²

h₁(δ, λ) := (δ C_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(δ)t whereδ∈(0,2) andλ(δ)is characterized by

λ(δ) := sup

λ >0 : h1(δ, λ) = 0, λm− δ−¹₄(2 +δ)λ >0

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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

2k(1−Π)Fk² kAFk ≤ 1

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

D[F]−λH[F]≥(λ_m−δ)X²+ δ λ_M 1 +λM

Y²−δ C_MX Y−2 +δ

4 λ(X²+Y²)

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Hypocoercivity

Corollary

For anyδ∈(0,2), ifλ(δ)is the largest positive root ofh1(δ, λ) = 0for whichλm−δ−¹₄(2 +δ)λ >0, then for any solutionF of the evolution equation

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

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Formal macroscopic (diffusion) limit

Scaled evolution equation εdF

dt +TF = 1 εLF

on the Hilbert spaceH. F_ε=F₀+ε F₁+ε²F₂+O(ε³) asε→0₊ ε⁻¹: LF0= 0,

ε⁰: TF0=LF1, ε¹: ^dF_dt⁰ +TF1=LF2

The first equation reads asu=F0= 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

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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=_dt^d u²₁+u²₂

=−2u²₂ macroscopic/diffusion limit: ^du_dt¹ =−k²u1

generalized entropy: H(u) =|u|²− _1+k^{δ k}2 u1u2

dH

dt = −

2− δ k² 1 +k²

u²₂− δ k²

1 +k²u²₁+ δ k 1 +k²u1u2

≤ −(2−δ)u²₂− δΛ

1 + Λu²₁+δ 2u1u2

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Plots for the toy problem

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1.0 u1²

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1.0 u1², u1²+u2²

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1.0 H

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1.0 1.2 D

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Some references

C. Mouhot and L. Neumann. Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus.

Nonlinearity, 19(4):969-998, 2006

F. Hérau. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot.

Anal., 46(3-4):349-359, 2006

J. Dolbeault, P. Markowich, D. Oelz, and C. Schmeiser. Non linear diffusions as limit of kinetic equations with relaxation collision kernels. Arch. Ration. Mech. Anal., 186(1):133-158, 2007.

J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for kinetic equations with linear relaxation terms. Comptes Rendus Mathématique, 347(9-10):511 - 516, 2009

J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. Transactions of the American Mathematical Society, 367(6):3807-3828, 2015

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The compact case

Fokker-Planck equation and scattering collision operators BA mode-by-mode (Fourier) hypocoercivity result

BEnlargement of the space by factorization

BApplication to the torus and numerical improvements Further results: Euclidean space with strong confinement

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

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Fokker-Planck and scattering collision operators

Two basic examples:

LinearFokker-Planckcollision operator Lf = ∆vf+∇v·(v f)

Linear relaxation operator (linear BGK) Lf =ρ(2π)^−d/2 exp(−|v|²/2)−f withρ=R

f dv

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Fokker-Planck equation with general equilibria

We consider the Cauchy problem

∂_tf+v· ∇xf =Lf , f(0, x, v) =f₀(x, v)

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 equilibriumM is positive, radially symmetric and

Z

R^d

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

lim

|v|→∞

|v|^k

γ(v) = lim

|v|→∞M(v)|v|^k= 0 ∀k∈(d,∞)

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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

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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 =M ρ_f−f , ρ_f(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

|M ρf−f|²dγ

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(Technicalities !) 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)² dv⁰dv

≥λ_m Z

R^d

(u−ρ_{u 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

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Euclidean space, confinement, Poincaré inequality

(H1) Regularity & Normalization: V ∈W_loc^2,∞(R^d),R

R^de^−Vdx= 1 (H2) Spectral gap condition (macroscopic Poincaré inequality): for

some Λ>0,∀u∈H¹(e^−Vdx) such thatR

R^du e^−Vdx= 0 R

R^d|u|²e^−Vdx≤ΛR

R^d|∇_xu|²e^−Vdx (H3) Pointwise conditions:

there existsc0>0,c1>0 andθ∈(0,1) s.t.

∆V ≤ ^θ₂|∇xV(x)|²+c0, |∇²_xV(x)| ≤c1(1 +|∇xV(x)|)∀x∈R^d (H4) Growth condition: R

R^d|∇_xV|²e^−Vdx <∞ Theorem (D., Mouhot, Schmeiser)

LetL be either a Fokker-Planck operator or a linear relaxation operator with a local equilibriumF(v) = (2π)^−d/2exp(−|v|²/2). Iff solves

∂tf+v· ∇xf− ∇xV · ∇vf =Lf then

∀t≥0, kf(t)−Fk²≤(1 +η)kf0−Fk²e^−λt

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Mode-by-mode decomposition

BSpectral decomposition (Hermite functions): linear Fokker-Planck operator

Lf = ∆vf+∇v·(v f) or the linear relaxation operator (linear BGK)

Lf =ρ(2π)^−d/2 exp(−|v|²/2)−f , withρ=R

f dv: (Arnold, Erb),(Achleitner, Arnold, Stürzer), (Achleitner, Arnold, Carlen),(Arnold, Einav, Wöhrer) BDecomposition inFourier modes

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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

fˆ(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)=

fˆ(t,·, v)

_L₂_(dµ(ξ)) The Cauchy problem is now decoupled in theξ-direction

∂_tfˆ+Tfˆ=Lf ,ˆ fˆ(0, ξ, v) = ˆf₀(ξ, v) Tfˆ=i(v·ξ) ˆf

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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=M ρ_F andTfˆ=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 = −i ξ·R

R^dv⁰F(v⁰)dv⁰ 1 + Θ|ξ|² M

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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

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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

The two “good” terms

− hLF, Fi ≥λmk(1−Π)Fk² hATΠF, Fi=AF = Θ|ξ|²

1 + Θ|ξ|²kΠFk² 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

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Mode-by-mode hypocoercivity with exponential weights

Theorem

Let us consider an admissibleM and a collision operatorL satisfying the assumptions, and takeξ∈R^d. Iffˆis a solution such that fˆ0(ξ,·)∈L²(dγ), then for any t≥0, we have

fˆ(t, ξ,·)

2 L²(dγ)

≤3e⁻^µ^ξ^t

fˆ₀(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

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Exponential convergence to equilibrium in T

^d

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

admissibleM, a collision operatorLsatisfying the assumptions. There exists a positive constantC such that the solutionf of the Cauchy problem onT^d×R^d with initial datumf0∈L²(dx dγ)satisfies

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

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Exponential convergence to equilibrium in T

^d

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 thatM is admissible,k∈(d,∞]and dγk:=γk(v)dv where γk(v) = 1 +|v|²^k/2

and k > d

There exists a positive constantCk such that the solutionf(t,·,·)on T^d×R^d with initial datumf0∈L²(dx dγk)satisfies

kf(t,·,·)−f_∞k_L2(dx dγk)≤Ck kf0−f_∞k_L2(dx dγk) e⁻¹⁴^Λ^t ∀t≥0 Tool: Enlargement of the space by thefactorization method

of(Gualdani, Mischler, Mouhot)

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Further references

Exponential rates in kinetic equations: (Talay 2001),(Wu 2001) In presence of a strongly confining potential(Hérau 2006 & 2007), (Mouhot, Neumann, 2006)

Hypo-elliptic methods(Hérau, Nier 2004),(Eckmann, Hairer, 2003),(Hörmander, 1967),(Kolmogorov, 1934),(Il’in, Has’min’ski, 1964)with applications to the Vlasov-Poisson-Fokker-Planck equation: (Victory, O’Dwyer, 1990),(Bouchut, 1993)

The non-compact case

The global picture:

Bby what can we replace themacroscopicPoincaré inequality ? Nash’s inequalityand a decay rate whenV = 0

Very weak confinement: Caffarelli-Kohn-Nirenberg inequalities and moments

With sub-exponential equilibria: weighted Poincaré inequalityor Hardy-Poincaré inequality

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Vlasov-Poisson-Fokker-Planck With sub-exponential local equilibria

A result based on Nash’s inequality

∂tf+v· ∇xf =Lf , (t, x, v)∈R⁺×R^d×R^d D[f]=−d

dtH[f]≥a

k(1−Π)fk²+ 2hATΠf, fi We observe that

A^∗f =TΠ 1 + (TΠ)^∗TΠ−1

f

=T 1 + (TΠ)^∗TΠ−1

Πf =MTuf =v M· ∇xuf

ifu_f is the solution in H¹(dx) ofu_f−Θ ∆_xu_f =ρ_f, and kuf(t,·)k_L1(dx)=kρf(t,·)k_L1(dx)=kf0k_L1(dx dv)

k∇xufk²_L2(dx)≤ 1

ΘhATΠf, fi kΠfk²≤ kufk²_L2(dx)+ 2hATΠf, fi

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Nash’s inequality kuk²_L2(dx)≤CNashkuk

4 d+2

L¹(dx)k∇uk

2d d+2

L²(dx) ∀u∈L¹∩H¹(R^d) UsekΠfk²≤Φ⁻¹ 2hATΠf, fi

with Φ⁻¹(y) :=y+ ^y_c_d+2^d to get k(1−Π)fk²+ 2hATΠf, fi≥Φ(kfk²)≥Φ _1+δ² H[f]

D[f]=−d

dtH[f]≥aΦ _1+δ² H[f]

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Algebraic decay rates in R

^d

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 orderk > d

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

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

(1 +t)⁻^d²

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The rate of decay of the heat equation

Ifρis a solution of the heat equation

∂ρ

∂t = ∆ρ with initial datumρ₀, then

kρ(t,·)k_L1(dx)=kρ₀k_L1(dx)

d

dtkρ(t,·)k²_L2(dx)=− k∇ρ(t,·)k²_L2(dx)≤ −Ckρ(t,·)k²⁺_L2(dx)⁴^d

by Nash’s inequality, and so

kρ(t,·)k²_L2(dx)=C kρ0k²_L2(dx)(1 +t)⁻^d²

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The global picture

Depending on the local equilibria and on the external potential (H1) and (H2) (which are Poincaré type inequalities) can be replaced by other functional inequalities:

Bmicroscopic coercivity (H1)

− hLF, Fi ≥λmk(1−Π)Fk²

=⇒weak Poincaré inequalitiesor Hardy-Poincaré inequalities Bmacroscopic coercivity (H2)

kTΠFk²≥λMkΠFk²

=⇒Nash inequality,weighted Nashor Caffarelli-Kohn-Nirenberg inequalities This can be done at the level of the diffusion equation

(homogeneous case) or at the level of thekinetic equation (non-homogeneous case)

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Diffusion (Fokker-Planck) equations

Potential

V

= 0

V

(x) = log

|x|

< d

V

(x) =

|x|^↵

↵2

(0, 1)

V

(x) =

|x|^↵

↵

1 Inequality Nash Ca↵arelli-Kohn -Nirenberg

Weak Poincar´e or Weighted Poincar´e

Poincar´e

Asymptotic behavior

t ^d/2

decay

t ^(d ^)/2

decay

t ^µ

or

t ^{2 (1}^k^↵)

convergence

e ^t

convergence

Table 1:

@_tu

=

u

+

r ·

(n

rV

)

Potential

V

= 0

V

(x) = log

|x|

< d

V

(x) =

|x|^↵

↵2

(0, 1)

V

(x) =

|x|^↵

↵

1 Inequality Nash Ca↵arelli-Kohn -Nirenberg

Weak Poincar´e or Weighted Poincar´e

Poincar´e

Asymptotic behavior

t ^d/2

decay

t ^(d ^)/2

decay

t ^µ

or

t ^{2 (1}^k^↵)

convergence

e ^t

convergence

Table 2:

@_tu

=

u

+

r ·

(n

rV

)

1

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Kinetic Fokker-Planck equations

B = Bouin, L = Lafleche, M = Mouhot, MM = Mischler, Mouhot S = Schmeiser

Potential V= 0 V(x) = log|x|

< d

V(x) =|x|^↵

↵2(0,1)

V(x) =|x|^↵

↵ 1, orT^d Macro Poincar´e

Micro Poincar´e F(v) =e ^hvi, 1

BDMMS:

t ^d/2 decay

BDS:t ^(d ^)/2 decay

Cao:e ^t^b, b <1, = 2 convergence

DMS, Mischler-

Mouhot e ^t convergence

F(v) =e ^hvi, 2(0,1)

BDLS:t ^⇣,

⇣= min ^d₂,^k}

decay

F(v) =hvi ^d

BDLS, fractional in progress

Table 1:@tf+v· rxf=Frv F ¹rvf . Notation:hvi=p 1 +|v|²

1

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Further references

Weak Poincaré inequality: (Röckner & Wang, 2001),(Kavian, Mischler),(Cao, PhD thesis), (Hu, Wang, 2019)+(Ben-Artzi, Einav) for recent spectral considerations

Weighted Nash inequalities: (Bakry, Bolley, Gentil, Maheux, 2012), (Wang, 2000, 2002, 2010)

Very weak confinement: Caffarelli-Kohn-Nirenberg

In collaboration with Emeric Bouin and Christian Schmeiser

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The macroscopic Fokker-Planck equation

∂u

∂t = ∆xu+∇x·(∇xV u) =∇x e^−V ∇x e^V u Herex∈R^d, d≥3, andV is a potential such thate^−V 6∈L¹(R^d) corresponding to avery weak confinement

Two examples

V₁(x) =γlog|x| and V₂(x) =γ loghxi withγ < dandhxi:=p

1 +|x|²for anyx∈R^d

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Abstract method and motivation The compact case The non-compact case Vlasov-Poisson-Fokker-Planck

Without confinement: Nash inequality The global picture

Very weak confinement: Caffarelli-Kohn-Nirenberg With sub-exponential local equilibria

V

= 0

< d ↵2

(0, 1)

↵

1 Inequality Nash Ca↵arelli-Kohn -Nirenberg

Weak Poincar´e or Weighted Poincar´e

Poincar´e

Asymptotic behavior

t ^d/2

decay

t ^(d ^)/2

decay

t ^µ

or

t ^{2 (1}^k^↵)

convergence

e ^t

convergence

Table 1:

@tu

=

u

+

r ·

(n

rV

)

Potential

V

= 0

V

(x) = log

|x|

< d

V

(x) =

|x|^↵

↵2

(0, 1)

V

(x) =

|x|^↵

↵

1 Inequality Nash Ca↵arelli-Kohn -Nirenberg

Weak Poincar´e or Weighted Poincar´e

Poincar´e

Asymptotic behavior

t ^d/2

decay

t ^(d ^)/2

decay

t ^µ

or

t ^{2 (1}^k^↵)

convergence

e ^t

convergence

Table 2:

@tu

=

u

+

r ·

(n

rV

)

1

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A first decay result

Theorem

Assume thatd≥3,γ <(d−2)/2 andV =V1 orV =V2

F any solutionuwith initial datumu0∈L¹₊∩L²(R^d), ku(t,·)k²₂≤ ku0k²₂

(1 +c t)^d² with c:= 4 d minn

1,1−_d−2²^γ o

C⁻¹_Nashku0k^4/d₂ ku0k^4/d₁ HereCNash denotes the optimal constant in Nash’s inequality

kuk²⁺₂ ⁴^d ≤CNash kuk₁⁴^dk∇uk²₂ ∀u∈L¹∩H¹(R^d)

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An extended range of exponents: with moments

Theorem

Letd≥1,0< γ < d,V =V₁ orV =V₂, andu₀∈L¹₊∩L² e^V

with

|x|^ku0

₁<∞for somek≥max{2, γ/2}

∀t≥0, ku(t,·)k²_L2(e^Vdx)≤ ku0k²_L2(e^Vdx) (1 +c t)⁻^d−γ² for somecdepending ond,γ,k,ku0k_L2(e^Vdx),ku0k₁, and

|x|^ku0

₁

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An extended range of exponents: in self-similar variables

u?(t, x) = c?

(1 + 2t)^d−γ² |x|^−γexp

− |x|² 2 (1 + 2t)

Here the initial data need have a sufficient decay...

c_? is chosen such thatku?k1=ku0k1

Theorem

Letd≥1,γ∈(0, d),V =V1 assume that

∀x∈R^d, 0≤u0(x)≤K u?(0, x) for some constantK >1

∀t≥0, ku(t,·)−u?(t,·)k_p≤K c¹⁻

1

? pku0k

1 p

1

e 2|γ|

^γ₂ 1−1 p

(1+2t)^−ζ^p for anyp∈[1,+∞), whereζ_p:= ^d₂ 1−¹_p

+₂¹_p min 2,_d−γ^d

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Proofs: basic case

d dt

Z

R^d

u²dx=−2 Z

R^d

|∇u|²dx+ Z

R^d

∆V |u|²dx with eitherV =V1or V =V2 and

∆V₁(x) =γd−2

|x|² and ∆V₂(x) =γ d−2

1 +|x|² + 2γ (1 +|x|²)² Forγ≤0: apply Nash’s inequality

d

dtkuk²₂≤ −2k∇uk²₂≤ − 2 CNash

ku0k^−4/d₁ kuk^2+4/d₂ For 0< γ <(d−2)/2: Hardy-Nash inequalities

Lemma

Letd≥3 andδ <(d−2)²/4 kuk²⁺₂ ⁴^d ≤Cδ

k∇uk²₂−δ Z

R^d

u²

|x|²dx

kuk₁⁴^d ∀u∈L¹∩ H¹(R^d) withCδ =CNash/

1−_(d−2)⁴^δ 2

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Proofs: moments

Growth of the moment

M_k(t) :=

Z

R^d

|x|^ku dx From the equation

M_k⁰ =k d+k−2−γ Z

R^d

u|x|^k−2dx≤k d+k−2−γ

M₀²^kM_k¹⁻²^k

then use the Caffarelli-Kohn-Nirenberg inequality Z

R^d

|x|^γu²dx≤CZ

R^d

|x|^−γ |∇(|x|^γu)|²dx ^aZ

R^d

|x|^k|u|dx 2(1−a)

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Proofs: self-similar solutions

The proof relies onuniform decay estimates+ Poincaré inequality in self-similar variables

Proposition

Letγ∈(0, d)and assume that 0≤u(0, x)≤c? σ+|x|²−γ/2

exp

−|x|² 2

∀x∈R^d withσ= 0 ifV =V₁ andσ= 1 ifV =V₂. Then

0≤u(t, x)≤ c?

(1 + 2t)^d−γ²

σ+|x|²−γ/2

exp

− |x|² 2 (1 + 2t)

for anyx∈R^d andt≥0

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The kinetic Fokker-Planck equation

Let us consider the kinetic equation

∂_tf+v· ∇_xf− ∇_xV · ∇_vf =Lf whereLf is one of the two following collision operators

(a) a Fokker-Planck operator Lf =∇v·

M∇v M⁻¹f (b) a scattering collision operator

Lf = Z

R^d

σ(·, v⁰) f(v⁰)M(·)−f(·)M(v⁰) dv⁰

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Potential V= 0 V(x) = log|x|

< d

V(x) =|x|^↵

↵2(0,1)

V(x) =|x|^↵

↵ 1, orT^d Macro Poincar´e

Micro Poincar´e F(v) =e ^h^vⁱ , 1

BDMMS:

t ^d/2 decay

BDS:t ^(d ^)/2 decay

Cao:e ^t^b, b <1, = 2 convergence

DMS, Mischler-

Mouhot e ^t convergence

F(v) =e ^h^vⁱ , 2(0,1)

BDLS:t ^⇣,

⇣= min ^d₂,^k}

decay

F(v) =hvi ^d

BDLS, fractional in progress

Table 2:@tf+v· rxf=Frv F ¹rvf . Notation:hvi=p 1 +|v|²

2

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Decay rates

∀(x, v)∈R^d×R^d, M(x, v) =M(v)e^−V^(x), M(v) = (2π)⁻^d² e⁻¹²^|v|² (H1) 1≤σ(v, v⁰)≤σ , ∀v , v⁰∈R^d, for some σ≥1 (H2)

Z

R^d

σ(v, v⁰)−σ(v⁰, v)

M(v⁰)dv⁰ = 0 ∀v∈R^d Theorem

Letd≥1,V =V2 with γ∈[0, d),k >max{2, γ/2} and f0∈L²(M⁻¹dx dv)such that

RR

R^d×R^dhxi^kf₀dx dv+RR

R^d×R^d|v|^kf₀dx dv <+∞

If(H1)–(H2) hold, then there existsC >0such that

∀t≥0, kf(t,·,·)k²_L2(M⁻¹dx dv)≤C(1 + t)⁻

d−γ 2

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With sub-exponential equilibria

BThe homogeneous Fokker-Planck equation with sub-exponential equilibriaF(v) =C_αe^−hvi^α,α∈(0,1)

– decay rates based on the weak Poincaré inequality(Kavian, Mischler)

– decay rates based on a weighted Poincaré / Hardy-Poincaré inequality

BThe kinetic Fokker-Planck equation with sub-exponential local equilibria andno confinement, the equation with linear scattering

In collaboration with Emeric Bouin, Laurent Lafleche and Christian Schmeiser

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Fokker-Planck with sub-exponential equilibria

We consider thehomogeneous Fokker-Planck equation

∂_tg=∇v·

F∇v F⁻¹g associated withsub-exponential equilibria

F(v) =Cαe^−hvi^α, α∈(0,1)

The corresponding Ornstein-Uhlenbeck equation forh=g/F is

∂th=F⁻¹∇v·

F∇vh

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Potential V = 0 V(x) = log|x|

< d

V(x) =|x|^↵

↵2(0,1)

V(x) =|x|^↵

↵ 1

Inequality Nash Ca↵arelli-Kohn -Nirenberg

Weak Poincar´e or Weighted Poincar´e

Poincar´e

Asymptotic behavior

t ^d/2 decay

t ^(d ^)/2 decay

t ^µort ^{2 (1}^k^↵) convergence

e ^t convergence Table 3:@tu= u+r ·(nrV)

2

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Weak Poincaré inequality

Z

R^d

h−eh

2

dξ≤Cα,τ

Z

R^d

|∇h|²dξ _1+τ^τ

h−eh

2 1+τ

L^∞(R^d)

for some explicit positive constantCα,τ,eh:=R

R^dhdξ. Using d

dt Z

R^d

h(t,·)−eh

2

dξ=−2 Z

R^d

|∇vh|²dξ whereh=g/F and dξ=Fdv + Hölder’s inequality R

R^d

h−eh

2

dξ≤

R

R^d

h−eh

2

hvi^−βdξ _τ+1^τ

R

R^d

h−eh

2 L^∞(R^d)

hvi^{β τ}dξ _1+τ¹

with (τ+ 1)/τ =β/η, then for with M= sup_s∈(0,t)

h(s,·)−eh

2/τ L^∞(R^d)

Z

R^d

h(t,·)−eh

2

dξ≤ Z

R^d

h(0,·)−eh

2

dξ −_τ¹

+ 2τ⁻¹ C^1+1/τα,τ Mt

!^−τ

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Weighted Poincaré inequality

There exists a constantC>0 such that Z

R^d

|∇h|²Fdv≥CZ

R^d

h−eh

2

hvi^−βFdv withβ= 2 (1−α),eh:=R

R^dh Fdv andF(v) =Cαe^−hvi^α and α∈(0,1)

Written in terms ofg=h F, the inequality becomes Z

R^d

∇_v F⁻¹g

2 F²dµ≥CZ

R^d

|g−g|² hvi^{−2 (1−α)}dµ where dµ=Fdv