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Hypocoercivity without confinement:

mode-by-mode analysis and decay rates in the Euclidean space

Jean Dolbeault

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

December 7, 2017

MFO workshop onClassical and Quantum Mechanical Models of Many-Particle Systems(3–8/12/2017)

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Outline

An abstract hypocoercivity result BAbstract statement

BA toy model

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

Decay rates in the Euclidean space without confinement BA direct approach

BIn collaboration with E. Bouin, S. Mischler, C. Mouhot, C. Schmeiser

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

References

E. Bouin, J. D., S. Mischler, C. Mouhot, C. Schmeiser.

Hypocoercivity without confinement. Preprint arXiv:1708.06180 E. Bouin, J. D., S. Mischler, C. Mouhot, C. Schmeiser. Two examples of accurate hypocoercivity estimates based on a mode-by-mode analysis in Fourier space. In preparation.

J. D., C. Mouhot, and C. Schmeiser. Hypocoercivity for kinetic equations with linear relaxation terms. Comptes Rendus

Mathématique, 347(9-10):511 – 516, 2009.

J. D., A. Klar, C. Mouhot, and C. Schmeiser. Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes. Applied Mathematics Research eXpress, 2012.

J. D., C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. Trans. Amer. Math. Soc., 367(6):3807–3828, 2015.

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

An abstract hypocoercivity result

BAbstract statement BA toy model

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

An abstract evolution equation

Let us consider the equation dF

dt +TF =LF (1)

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)−1

(TΠ)

denotes the adjoint with respect toh·,·i

Π is the orthogonal projection onto the null space ofL

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

The assumptions

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

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

kTΠFk2λMkΠFk2 (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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

Exponential decay of the entropy

λ= 3 (1+λλM

M)minn

1, λm,(1+λλmλM

M)CM2

o

,δ=12 min

1, λm,(1+λλmλM

M)CM2

h1(δ, λ) := (δ CM)2−4

λmδ−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, λmδ14(2 +δ)λ >0

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

Sketch of the proof

SinceATΠ = 1 + (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)G 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]≥(λmδ)X2+ δ λM

1 +λM

Y2δ CMX Y−2 +δ

4 λ(X2+Y2)

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

Hypocoercivity

Corollary

For anyδ∈(0,2), ifλ(δ)is the largest positive root ofh1(δ, λ) = 0for whichλmδ14(2 +δ)λ >0, then for any solutionF of (1)

kF(t)k2≤ 2 +δ

2− δe−λ(δ)tkF(0)k2t≥0 From the norm equivalence ofH[F] andkFk2

2−δ

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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

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: dtd|u|2=−2u22

macroscopic limit: dudt1 =−k2u1

generalized entropy: H(u) =|u|21+kδ k2 u1u2 dH

dt = −

2− δ k2 1 +k2

u22δ k2

1 +k2u21+ δ k 1 +k2u1u2

≤ −(2−δ)u22δΛ

1 + Λu21+δ 2u1u2

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Strategy Result and proof Toy model

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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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)

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Fokker-Planck equation with general equilibria

We consider the Cauchy problem

tf+v· ∇xf =Lf , f(0, x, v) =f0(x, v) (2) for a distribution functionf(t, x, v), withpositionvariablex∈Rd or x∈Td the flat d-dimensional torus

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

Mv M−1fi

Notation and assumptions: anadmissible local equilibriumM is positive, radially symmetric and

Z

Rd

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

lim

|v|→∞

|v|k

γ(v) = lim

|v|→∞M(v)|v|k= 0 ∀k∈(d,∞)

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Definitions

Θ = 1 d

Z

Rd

|v|2M(v)dv= Z

Rd

(v·e)2M(v)dv for an arbitrarye∈Sd−1

Z

Rd

vv M(v)dv= Θ Id Then

θ= 1

d k∇vMk2L2(dγ)=4 d

Z

Rd

|∇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

Rd

|∇u|2M dvλm

Z

Rd

u

Z

Rd

u M dv 2

M dv

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Scattering collision operators

Scatteringcollision operator Lf =

Z

Rd

σ(·, v0) f(v0)M(·)−f(·)M(v0) dv0

Main assumption on thescattering rateσ: for some positive, finiteσ 1≤σ(v, v0)≤σv, v0∈Rd

Example: linear BGK operator

Lf =M ρff , ρf(t, x) = Z

Rd

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

Z

Rd

Lf dv= 0 and we have

Z

Rd

|Lf|2≤4σ2 Z

Rd

|M ρff|2

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

The symmetry condition Z

Rd

σ(v, v0)−σ(v0, v)

M(v0)dv0 = 0 ∀v∈Rd implies thelocal mass conservationR

RdLf 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

Rd×Rd

σ(v, v0)M(v)M(v0) (u(v)−u(v0))2dv0dv

λm

Z

Rd

(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, thenλm= 1

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Fourier modes

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

f(t, x, v) = Z

Rd

fˆ(t, ξ, v)e−i x·ξdµ(ξ)

dµ(ξ) = (2π)−d and is the Lesbesgue measure ifx∈Rd dµ(ξ) = (2π)−d P

z∈Zdδ(ξz) is discrete forx∈Td

Parseval’s identity ifξ∈Zd and Plancherel’s formula ifx∈Rd read kf(t,·, v)kL2(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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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

Z

Rd

|F|2dγ , ΠF =M Z

Rd

F dv=M ρF

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

Z

Rd

|v·ξ|2M(v)dv = Θ|ξ|2F|2= Θ|ξ|2kΠFk2 (H2)Macroscopic coercivitykTΠFk2λMkΠFk2: λM = Θ|ξ|2 (H3)R

Rdv M(v)dv= 0 The operatorAis given by

AF = −i ξ·R

Rdv0F(v0)dv0 1 + Θ|ξ|2 M

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

A mode-by-mode hypocoercivity result

kAFk=kA(1−Π)Fk ≤ 1 1 + Θ|ξ|2

Z

Rd

|(1−Π)F|

M |v·ξ|M dv

≤ 1

1 + Θ|ξ|2k(1−Π)Fk Z

Rd

(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

Rd

|(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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

TAF(v) =−(v·ξ)M 1 + Θ|ξ|2

Z

Rd

(v0·ξ) (1−Π)F(v0)dv0 is estimated by

kTAFk ≤ Θ|ξ|2

1 + Θ|ξ|2k(1−Π)Fk (H4) holds withCM = κ1+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 +λM)CM2 =Θ 1 + Θ|ξ|2

(κ+ Θ|ξ|)2 ≥ Θ κ2+ Θ

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Mode-by-mode hypocoercivity with exponential weights

Theorem

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

fˆ(t, ξ,·)

2

L2(dγ)≤3eµξt

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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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 · k1c1k · k2. LetB andA+Bbe the generators of the strongly continuous semigroupseBt ande(A+B)ton B1. If for all t≥0,

e(A+B)t

2→2c2e−λ2t, eBt

1→1c3e−λ1t, kAk1→2c4

wherek · ki→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 +|λ1λ2|−1

emin{λ12}t forλ16=λ2 C(1 +t)e−λ1t forλ1=λ2

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Integrating the identity dsd e(A+B)seB(t−s)

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

e(A+B)t=eBt+ Z t

0

e(A+B)sAeB(t−s)ds

The proof is completed by the straightforward computation e(A+B)t

1→1c3e−λ1t+c1 Z t

0

e(A+B)sAeB(t−s) 1→2ds

c3e−λ1t+c1c2c3c4e−λ1t Z t

0

e1−λ2)sds

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Weights with polynomial growth

Let us consider the measure

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

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

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

LetΛ = 3 max{1,Θ}Θ minn 1,κλ2mΘ

o

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

fˆ(t, ξ,·)

2

L2(dγk)C eµξt

fˆ0(ξ,·)

2

L2(dγk)t≥0

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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

Rd

(L−A)(F)F dγk ≤ −λ1

Z

Rd

F2k

for someλ1>0 ifk > d, andλ2=µξ/2≤1/4 Scattering operator:

AF(v) = M(v) Z

Rd

σ(v, v0)F(v0)dv0 BF(v) = −

i(v·ξ) + Z

Rd

σ(v, v0)M(v0)dv0

F(v) Boundedness: kAFkL2(dγ)σ R

Rdγk−1dv1/2

kFkL2(dγk)

λ1= 1 andλ2=µξ/2≤1/4

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Exponential convergence to equilibrium in T

d

The unique global equilibrium in the casex∈Td is given by f(x, v) =ρM(v) with ρ= 1

|Td| Z Z

Td×Rd

f0dx dv

Theorem

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

We consider an admissibleM, a collision operatorLsatisfying Assumption(H), andΛgiven by (3)

There exists a positive constantCk such that the solutionf of (2) on Td×Rd with initial datumf0∈L2(dx dγk)satisfies

kf(t,·,·)−fkL2(dx dγk)Ck kf0fkL2(dx dγk) e14Λtt≥0

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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

fˆ(t,0,·)−fˆ(0,·)

L2(dγ)

fˆ0(0,·)−fˆ(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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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) = e12|v|2 (2π)d/2 andρf :=R

Rdf dv is the spatial density

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

Where do we have space for improvements ?

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

≥(λmδ)X2+ δ λM

1 +λM Y2δ CMX Yλ

2 X2+Y2+δ X Y

≥(λmδ)X2+ δ λM

1 +λM Y2δ CMX Y −2 +δ

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

mδ)X2+ δ λM

1 +λM Y2δ CMX Yλ

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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

2 4 6 8 10

0.1 0.2 0.3 0.4

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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

mδ)X2+ δ λM

1 +λM Y2δ CMX Yλ

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

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

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

2 4 6 8 10

0.1 0.2 0.3 0.4 0.5

J. Dolbeault Hypocoercivity without confinement

(34)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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

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

AF = −i ξ·R

Rdv0F(v0)dv0 1 +|ξ|2 M

We can consider

AεF = −i ξ·R

Rdv0F(v0)dv0 ε+|ξ|2 M and look for the optimal value ofε...

J. Dolbeault Hypocoercivity without confinement

(35)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

2 4 6 8 10

0.1 0.2 0.3 0.4 0.5

J. Dolbeault Hypocoercivity without confinement

(36)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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

A0F= −i ξ·R

Rdv0F(v0)dv0

|ξ|2 M

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

2 4 6 8 10

0.05 0.10 0.15 0.20 0.25 0.30 0.35

J. Dolbeault Hypocoercivity without confinement

(37)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

2 4 6 8 10

0.1 0.2 0.3 0.4

J. Dolbeault Hypocoercivity without confinement

(38)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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

kf(t,·,·)−fk2L2(dx dγ)≤C?kf0k2L2(dx dγ)eλ?tt≥0 withf(x, v) =M(v)RR

Td×Rdf0(x, v)dx dv C?≈1.75863 andλ?= 132(5−2√

3)≈0.236292.

Warning: work in progress

J. Dolbeault Hypocoercivity without confinement

(39)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Two collision kernels and Fourier modes Hypocoercivity results

Application to the torus and some numerical results

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, Stürzer)

(Arnold, Erb)

(Arnold, Einav, Wöhrer)

J. Dolbeault Hypocoercivity without confinement

(40)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Algebraic decay rates inRd

Improved decay rate for zero average solutions

Decay rates in the whole space

J. Dolbeault Hypocoercivity without confinement

(41)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Algebraic decay rates inRd

Improved decay rate for zero average solutions

Algebraic decay rates in R

d

On the whole Euclidean space, we can define the entropy H[f] := 12kfk2L2(dx dγk)+δhAf, fidx dγk

Replacing themacroscopic coercivitycondition byNash’s inequality kuk2L2(dx)≤CNashkuk

4 d+2

L1(dx)k∇uk

2d d+2

L2(dx)

proves that

H[f]≤C

H[f0] +kf0k2L1(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,·,·)k2L2(dx dγk)C

kf0k2L2(dx dγk)+kf0k2L2(dγk; L1(dx))

(1 +t)d2

J. Dolbeault Hypocoercivity without confinement

(42)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Algebraic decay rates inRd

Improved decay rate for zero average solutions

A direct proof... Recall thatµξ= 1+|ξ|Λ|ξ|22

By the Plancherel formula kf(t,·,·)k2L2(dx dγk)C

Z

Rd

Z

Rd

eµξt|fˆ0|2

k

if|ξ|<1, thenµξΛ2 |ξ|2 Z

|ξ|≤1

eµξt|fˆ0|2C kf0(·, v)k2L1(dx)

Z

Rd

eΛ2|ξ|2t

C kf0(·, v)k2L1(dx)td2 if|ξ| ≥1, thenµξ ≥Λ/2 when |ξ| ≥1

Z

|ξ|>1

eµξt|fˆ0|2C eΛ2t kf0(·, v)k2L2(dx)

J. Dolbeault Hypocoercivity without confinement

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An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Algebraic decay rates inRd

Improved decay rate for zero average solutions

Improved decay rate for zero average solutions

Theorem

Assume thatf0∈L1loc(Rd×Rd)with RR

Rd×Rdf0(x, v)dx dv= 0 and C0:=kf0k2L2(dγk+2; L1(dx))+kf0k2L2(dγk; L1(|x|dx))+kf0k2L2(dx dγk)<Then there exists a constantck>0 such that

kf(t,·,·)k2L2(dx dγk)ck C0(1 +t)(1+d2)

J. Dolbeault Hypocoercivity without confinement

(44)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Algebraic decay rates inRd

Improved decay rate for zero average solutions

Step 1: Decay of the average in space, factorization

x-average in space

f(t, v) :=

Z

Rd

f(t, x, v)dx withR

Rdf(t, v)dv= 0 and observe thatf solves aFokker-Planck equation

tf=Lf

From themicroscopic coercivity property, we deduce that kf(t,·)k2L2(dγ)≤ kf(0,·)k2L2(dγ)e−λmt Factorisation

kf(t,·)k2L2(|v|2k)C kf0k2L2(|v|2k;L1(dx)) e−λ t

J. Dolbeault Hypocoercivity without confinement

(45)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Algebraic decay rates inRd

Improved decay rate for zero average solutions

Step 2: Improved decay of f

Let us defineg:=ffϕ, withϕ(x) := (2π)−d/2e−|x|2/2 The Fourier transform ˆgsolves

tˆg+Tˆg=LˆgfTϕˆwithTϕˆ=i(v·ξ) ˆϕ Duhamel’s formula

ˆ

g= ei(L−T)tgˆ0

| {z }

C e12µξ tg0(ξ,·)kL2(dγk)

+ Z t

0

ei(L−T) (t−s)(−f(s, v)Tϕ(ξ))ˆ

| {z }

C e

µξ 2 (t−s)

kf(s,·)kL2(|v|2dγk)|ξ| |ϕ(ξ)|ˆ

ds

ˆ

g0(ξ, v) = ˆg0(0, v)

| {z }

=0

+R|ξ|

0 ξ

|ξ|· ∇ξˆg0 η |ξ|ξ , v

yields

g0(ξ, v)| ≤ |ξ| k∇ξgˆ0(·, v)kL(dv)≤ |ξ| kg0(·, v)kL1(|x|dx)

µξ = Λ|ξ|2/(1 +|ξ|2)≥Λ/2 if|ξ|>1 (contribution O(eΛ2t)) and R

|ξ|≤1

R

Rd

ei(L−T)tˆg0

2k≤ Z

Rd

|ξ|2eΛ2|ξ|2t

| {z }

=O(t−(d+2))

kg0k2L2(dγk; L1(|x|dx))

J. Dolbeault Hypocoercivity without confinement

(46)

An abstract hypocoercivity result Mode-by-mode hypocoercivity Decay rates in the whole space

Algebraic decay rates inRd

Improved decay rate for zero average solutions

These slides can be found at

http://www.ceremade.dauphine.fr/∼dolbeaul/Lectures/

BLectures The papers can be found at

http://www.ceremade.dauphine.fr/∼dolbeaul/Preprints/

BPreprints / papers

For final versions, useDolbeaultas login andJeanas password

Thank you for your attention !

J. Dolbeault Hypocoercivity without confinement

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