From ϕ-entropies to hypocoercivity without conﬁnement

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From ϕ-entropies to hypocoercivity without confinement

Jean Dolbeault

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

June 26, 2018

An Analyst, a Geometer & a Probabilist Walk Into a Bar (Cardiff University, 25-29 June 2018)

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Fromϕ-entropies to H1 hypocoercivity An abstract hypocoercivity result Mode-by-mode hypocoercivity

Outline

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

Bϕ-hypocoercivity (H¹ framework) An L² abstract hypocoercivity result BAbstract statement

BA toy model

L² framework: 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

BA more numerical point of view

BDecay rates in the Euclidean space without confinement BIn collaboration with X. Li

and E. Bouin, S. Mischler, C. Mouhot, C. Schmeiser

J. Dolbeault Fromϕ-entropies to hypocoercivity without confinement

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ϕ-entropies and diffusions ϕ-hypocoercivity (H1 framework)

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)

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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) :=wlogw−(w−1) The invariant measure

dγ=e^−ψdx

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

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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,·)dγand limt→+∞w(t,·) = 1 d

dtE[w] =− Z

R^d

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

I[w]≥ΛE[w] ∀w∈H¹(R^d, dγ) (entropy – entropy production inequality), then

E[w(t,·)]≤E[w0]e^−Λ^t ∀t≥0 Fokker-Planck equation: u=w γ converges tou_?=γ

∂u

∂t = ∆u+∇x·(u∇xψ)

A nonnegative solution with initial datumu0∈L¹(R^d, dx) and R

R^du0dx=M >0 has constant massM =R

R^du(t,·)dxfor anyt >0, and converges towards the unique stationary solution

u_?=M e^−ψ R

R^de^−ψdx

Without loss of generality, we shall assume thatM = 1. Then one observes thatw=u/u_? solves (??), which allows to control the rate of convergence ofutou?. A list of results concerning the solutions

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Generalized Csiszár-Kullback-Pinsker inequality

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

Proposition

Letp∈[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. IfA:= inf_s∈(0,∞)s^2−pϕ⁰⁰(s)>0, then

E[w]≥2⁻²^pAminn

1,kwk^p−2_Lp(R^d,dγ)

okw−1k²_Lp(R^d,dγ)

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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

Ifdγ1 anddγ2 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)

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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 aC² function such that ϕ⁰⁰>0 andwe:=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

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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 ws.t.

Z

Ω

w dγ= 1 (Bakry, Emery 1985)

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An interpolation inequality

Corollary

Assume thatq∈[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)

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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 withκp= (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 thatq∈(1,2) anddγ= (2π)^−d/2e^−|x|²^/2dx. There exists a strictly convex functionF such thatF(0) = 0 andF⁰(0) = 1 and

F

kfk²_L2(R^d,dγ)−1

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

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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

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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⁻¹²^(|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

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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

ϕ_p(g)dµ Proposition

Letp∈[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 ratee^−t is sharp ast→+∞

(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,·)]

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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

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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

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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

Letp∈(1,2) andhbe a solution of the kinetic Ornstein-Uhlenbeck equation. Then there exists a functionλ:R⁺→[1/2,1)such that λ(0) = limt→+∞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 anyt≥0we have the global estimate Jλ(t)[h(t,·)]≤J1/2[h0] exp

−2 Z t

0

ρ(s)ds

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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

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Strategy Result and proof Toy model

An abstract hypocoercivity result

BAbstract statement BA toy model

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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

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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≥λMkΠFk² (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 ≤ −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

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] 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 CM)²−4

λm− δ−2 +δ

4 λ δ λM

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

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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 +λM

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 By the Cauchy-Schwarz inequality, for anyµ >0

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]≥(λm−δ)X²+ δ λM

1 +λM

Y²−δ CMX 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 (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)

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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=−2u²₂

macroscopic 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 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

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Hypocoercivity results

Application to the torus and some numerical results Decay rates in the whole space

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)

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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) (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 equilibriumM 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,∞)

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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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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

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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 ρF

andTfˆ=i(v·ξ) ˆf,TΠF=i(v·ξ)ρFM, kTΠFk²=|ρF|²

Z

R^d

|v·ξ|²M(v)dv = Θ|ξ|²|ρF|²= Θ|ξ|²kΠFk² (H2)Macroscopic coercivitykTΠFk²≥λMkΠFk²: λM = 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

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 Assumption(H), 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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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. LetB andA+Bbe the generators of the strongly continuous semigroupse^Bt ande^(A+B)^ton B1. If for all t≥0,

e^(A+B)t

2→2≤c2e^−λ²^t, e^Bt

_1→1≤c3e^−λ¹^t, kAk_1→2≤c4

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

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

e^(A+B)^t _1→1≤

( C 1 +|λ1−λ₂|⁻¹

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

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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^B^t+ 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^−λ¹^t+c₁ Z t

0

e^(A+B)^sAe^B(t−s) 1→2ds

≤c₃e^−λ¹^t+c₁c₂c₃c₄e^−λ¹^t Z t

0

e^(λ¹^−λ²⁾^sds

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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|²^k/2 for an arbitraryk∈(d,+∞)

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

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

o

andk∈(d,∞]. For anyξ∈R^d if fˆis a solution with initial datum fˆ₀(ξ,·)∈L²(dγ_k), then there exists a constantC=C(k, d, σ)such that

fˆ(t, ξ,·)

2

L²(dγ_k)≤C e⁻^µ^ξ^t

fˆ0(ξ,·)

2

L²(dγ_k) ∀t≥0

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Fokker-Planck: AF =N χRF andBF=−i(v·ξ)F+LF−AF N andR are two positive constants,χ is a smooth cut-off function andχR:=χ(·/R)

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

Z

R^d

(L−A)(F)F dγk ≤ −λ1

Z

R^d

F²dγk

for someλ₁>0 ifk > d, andλ₂=µ_ξ/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_k⁻¹dv^1/2

kFk_L2(dγk)

λ1= 1 andλ2=µξ/2≤1/4

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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 thatk∈(d,∞]andγ has an exponential growth ifk=∞.

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

There exists a positive constantCk such that the solutionf of (3) 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

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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

fˆ(t,0,·)−fˆ_∞(0,·)

_L₂_(dγ)≤

fˆ₀(0,·)−fˆ_∞(0,·)

_L₂_(dγ) e^−t Otherwiseµ_ξ ≥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

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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 =ρ_fM 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

From ϕ-entropies to hypocoercivity without conﬁnement

From ϕ-entropies to hypocoercivity without confinement

Outline

From ϕ-entropies to H 1 hypocoercivity

Definition of the ϕ-entropies

Diffusions

Generalized Csiszár-Kullback-Pinsker inequality

Convexity, tensorization and sub-additivity

Perturbation (Holley-Stroock type) results

Entropy – entropy production inequalities, linear flows

An interpolation inequality

Improved entropy – entropy production inequalities

ϕ-hypocoercivity (H 1 framework)

Sharp rates for the kinetic Fokker-Planck equation

Improved rates (in the large entropy regime)

An abstract hypocoercivity result

An abstract evolution equation

The assumptions

Equivalence and entropy decay

Exponential decay of the entropy

Sketch of the proof

Hypocoercivity

A toy problem

Plots for the toy problem

Mode-by-mode hypocoercivity

Fokker-Planck equation with general equilibria

Scattering collision operators

Fourier modes

A mode-by-mode hypocoercivity result

Mode-by-mode hypocoercivity with exponential weights

Enlargement of the space by factorization

Weights with polynomial growth

Exponential convergence to equilibrium in T

Computation of the constants

From ϕ-entropies to H ¹ hypocoercivity

ϕ-hypocoercivity (H ¹ framework)