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)
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
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
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
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)∗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
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:
kTΠFk2≥λMkΠFk2 (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
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
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
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
J. Dolbeault Hypocoercivity without confinement
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)∗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)
J. Dolbeault Hypocoercivity without confinement
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 whichλm−δ−14(2 +δ)λ >0, then for any solutionF of (1)
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 2−4δkF0k2≤H[F0] so thatλ?≥supδ∈(0,2)λ(δ)
J. Dolbeault Hypocoercivity without confinement
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|2− 1+kδ k2 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
J. Dolbeault Hypocoercivity without confinement
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
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
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
M∇v M−1fi
Notation and assumptions: anadmissible local equilibriumM is positive, radially symmetric and
Z
Rd
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,∞)
J. Dolbeault Hypocoercivity without confinement
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
v⊗v 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 dv≥λm
Z
Rd
u−
Z
Rd
u M dv 2
M dv
J. Dolbeault Hypocoercivity without confinement
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)≤σ ∀v, v0∈Rd
Example: linear BGK operator
Lf =M ρf−f , ρf(t, x) = Z
Rd
f(t, x, v)dv Local mass conservation
Z
Rd
Lf dv= 0 and we have
Z
Rd
|Lf|2dγ≤4σ2 Z
Rd
|M ρf−f|2dγ
J. Dolbeault Hypocoercivity without confinement
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
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)−ddξ anddξ is the Lesbesgue measure ifx∈Rd dµ(ξ) = (2π)−d P
z∈Zdδ(ξ−z) 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
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|2|ρF|2= Θ|ξ|2kΠFk2 (H2)Macroscopic coercivitykTΠFk2≥λMkΠFk2: 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
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
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
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)≤3e−µξt
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
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 · k1≤c1k · 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→2≤c2e−λ2t, eBt
1→1≤c3e−λ1t, kAk1→2≤c4
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 +|λ1−λ2|−1
e−min{λ1,λ2}t forλ16=λ2 C(1 +t)e−λ1t forλ1=λ2
J. Dolbeault Hypocoercivity without confinement
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→1≤c3e−λ1t+c1 Z t
0
e(A+B)sAeB(t−s) 1→2ds
≤c3e−λ1t+c1c2c3c4e−λ1t Z t
0
e(λ1−λ2)sds
J. Dolbeault Hypocoercivity without confinement
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
dγk :=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,κλ2m+ΘΘ
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 e−µξt
fˆ0(ξ,·)
2
L2(dγk) ∀t≥0
J. Dolbeault Hypocoercivity without confinement
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
F2dγk
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
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
dThe unique global equilibrium in the casex∈Td is given by f∞(x, v) =ρ∞M(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,·,·)−f∞kL2(dx dγk)≤Ck kf0−f∞kL2(dx dγk) e−14Λt ∀t≥0
J. Dolbeault Hypocoercivity without confinement
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
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) = e−12|v|2 (2π)d/2 andρf :=R
Rdf dv is the spatial density
J. Dolbeault Hypocoercivity without confinement
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]
≥(λm− δ)X2+ δ λM
1 +λM Y2−δ CMX Y −λ
2 X2+Y2+δ X Y
≥(λm−δ)X2+ δ λM
1 +λM Y2−δ CMX 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−δ CMX Y −λ
2 X2+Y2+δ X Y λm= 1,λM =|ξ|2andCM =|ξ|(1 +|ξ|)/(1 +|ξ|2)
J. Dolbeault Hypocoercivity without confinement
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
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
(λm− δ)X2+ δ λM
1 +λM Y2−δ CMX Y −λ
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
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
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
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
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
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
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,·,·)−f∞k2L2(dx dγ)≤C?kf0k2L2(dx dγ)e−λ?t ∀t≥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
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
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
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
dOn 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
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
e−µξt|fˆ0|2dξ
dγk
if|ξ|<1, thenµξ ≥Λ2 |ξ|2 Z
|ξ|≤1
e−µξt|fˆ0|2dξ≤C kf0(·, v)k2L1(dx)
Z
Rd
e−Λ2|ξ|2tdξ
≤C kf0(·, v)k2L1(dx)t−d2 if|ξ| ≥1, thenµξ ≥Λ/2 when |ξ| ≥1
Z
|ξ|>1
e−µξt|fˆ0|2dξ≤C e−Λ2t kf0(·, v)k2L2(dx)
J. Dolbeault Hypocoercivity without confinement
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
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|2dγk)≤C kf0k2L2(|v|2dγk;L1(dx)) e−λ t
J. Dolbeault Hypocoercivity without confinement
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:=f −f•ϕ, withϕ(x) := (2π)−d/2e−|x|2/2 The Fourier transform ˆgsolves
∂tˆg+Tˆg=Lˆg−f•TϕˆwithTϕˆ=i(v·ξ) ˆϕ Duhamel’s formula
ˆ
g= ei(L−T)tgˆ0
| {z }
C e−12µξ tkˆg0(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
dη yields
|ˆg0(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(e−Λ2t)) and R
|ξ|≤1
R
Rd
ei(L−T)tˆg0
2dγkdξ≤ Z
Rd
|ξ|2e−Λ2|ξ|2tdξ
| {z }
=O(t−(d+2))
kg0k2L2(dγk; L1(|x|dx))
J. Dolbeault Hypocoercivity without confinement
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
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J. Dolbeault Hypocoercivity without confinement