Emeric Bouin, Jean Dolbeault, Stéphane Mischler, Clément Mouhot & Christian Schmeiser

by

Emeric Bouin, Jean Dolbeault, Stéphane Mischler, Clément Mouhot & Christian Schmeiser

Abstract. —In this paper, hypocoercivity methods are applied to linear kinetic equa- tions with mass conservation and without confinement, in order to prove that the solutions have an algebraic decay rate in the long-time range, which the same as the rate of the heat equation. Two alternative approaches are developed: an analy- sis based on decoupled Fourier modes and a direct approach where, instead of the Poincaré inequality for the Dirichlet form, Nash’s inequality is employed. The first approach is also used to provide a simple proof of exponential decay to equilibrium on the flat torus. The results are obtained on a space with exponential weights and then extended to larger function spaces by a factorization method. The optimality of the rates is discussed. Algebraic rates of decay on the whole space are improved when the initial datum has moment cancellations.

1. Introduction We consider the Cauchy problem

(1) ∂tf+v· ∇xf =

L

f, f(0,x,v)=f₀(x,v)

for a distribution functionf(t,x,v), withpositionvariablex∈R^d,velocityvariable v∈R^d, and withtime t≥0. Concerning thecollision operator

L

, we shall consider two cases:

(a) Fokker-Planckcollision operator:

L

f = ∇v·h M∇v¡

M⁻¹f¢i , (b) Scatteringcollision operator:

L

f = Z

R^dσ(·,v⁰)¡

f(v⁰)M(·)−f(·)M(v⁰)¢ d v⁰.

2010Mathematics Subject Classification. — Primary: 82C40. Secondary: 76P05; 35H10; 35K65;

35P15; 35Q84.

Key words and phrases. — Hypocoercivity; linear kinetic equations; Fokker-Planck operator; scat- tering operator; transport operator; Fourier mode decomposition; Nash’s inequality; factorization method; Green’s function; micro/macro decomposition; diffusion limit.

Corresponding author: Émeric Bouin.

We shall make the following assumptions on thelocal equilibrium M(v) and on thescattering rateσ(v,v⁰):

Z

R^dM(v)d v=1 , ∇v

pM∈L²(R^d) , M∈C(R^d) , (H1)

M=M(|v|) , 0<M(v)≤c₁e^−c2|v|, ∀v∈R^d, for somec₁,c₂>0 . 1≤σ(v,v⁰)≤σ, ∀v,v⁰∈R^d, for someσ≥1 .

(H2) Z

R^d

¡σ(v,v⁰)−σ(v⁰,v)¢

M(v⁰)d v⁰=0, ∀v∈R^d. (H3)

Before stating our main results, let us list some preliminary observations.

(i) A typical example of alocal equilibriumsatisfying (H1) is the Gaussian

(2) M(v)= e^−|v|

2 2

(2π)^d/2.

(ii) Withσ≡1, Case (b) includes the relaxation operator

L

f =Mρf−f, also known as thelinear BGK operator, with position density defined by

ρf(t,x) :=

Z

R^df(t,x,v)d v.

(iii) Positivity and exponential decay of the local equilibrium are essential for our approach. The assumption on the gradient and continuity are technical and only needed for some of our results. Rotational symmetry is not important, but as- sumed for computational convenience. However the property

Z

R^dv M(v)d v=0 , i.e.,zero flux in local equilibrium, is essential.

(iv) Since micro-reversibility (or detailed balance),i.e., symmetry ofσ, is not re- quired, Assumption (H3) is needed formass conservation,i.e.,

Z

R^d

L

f d v=0 ,

in Case (b). The boundedness away from zero ofσin (H2) guarantees coercivity of

L

relative to its nullspace (such bound can always be writtenσ≥1 by scaling).

Sincee^tLpropagates probability densities,i.e., conserves mass and nonnegativ- ity,

L

dissipates convex relative entropies, implying in particular

Z

R^d

L

f f

Md v≤0 .

This suggests to use the L²-space with the measuredγ∞:=γ∞d v, whereγ∞(v)= M(v)⁻¹, as a functional analytic framework (the subscript∞will make sense later).

We shall need themicroscopic coercivityproperty

(H4) −

Z

R^df

L

f dγ∞≥λm

Z

R^d

¡f −Mρf

¢2

dγ∞,

with some λm >0. In Case (a) it is equivalent to the Poincaré inequality with weightM,

Z

R^d|∇vh|²M d v≥λm

Z

R^d

µ h−

Z

R^dh M d v

¶2

M d v,

for allh=f/M∈H¹(M d v). It holds as a consequence of the exponential decay assumption in (H1) (see,e.g., [29,2]). For the normalized Gaussian (2) the optimal constant is known to beλm =1 (see for instance [4] and references therein). In

Case (b), (H4) means 1

2 Ï

R^d×R^dσ(v,v⁰)M(v)M(v⁰)¡

u(v)−u(v⁰)¢2

d v⁰d v≥λm

Z

R^d

¡u−ρu M¢2

M d v, for allu=f/M∈L²(M d v), and it holds withλm=1 as a consequence of the lower bound forσin Assumption (H2).

Although the transport operator does not contribute to entropy dissipation, its dispersion in thex-direction in combination with the dissipative properties of the collision operator yields the desired decay results. In order to perform amode- by-mode hypocoercivityanalysis, we introduce the Fourier representation with re- spect tox,

f(t,x,v)= Z

R^d

fˆ(t,ξ,v)e^{+i x·ξ}dµ(ξ) ,

wheredµ(ξ)=(2π)^−ddξanddξis the Lesbesgue measure onR^d. The normaliza- tion ofdµ(ξ) is chosen such that Plancherel’s formula reads

°

°f(t,·,v)°

°_{L2(d x)}=°

°fˆ(t,·,v)°

°_L2(dµ(ξ))

with a straightforward abuse of notations. The Cauchy problem (1) in Fourier variables is now decoupled in theξ-direction:

(3) ∂tfˆ+i(v·ξ) ˆf =

L

f^ˆ, fˆ(0,ξ,v)=fˆ₀(ξ,v) .

Our main results are devoted tohypocoercivity without confinement: when the variablexis taken inR^d, we assume that there is no potential preventing the run- away corresponding to|x| → +∞. So far, hypocoercivity results have been ob- tained either in the compact case corresponding to a bounded domain inx, for instanceT^d, or in the whole Euclidean space with an external potentialV such that the measuree^−Vd xadmits a Poincaré inequality. Usually other technical as- sumptions are required onVand there are many variants (for instance one can as- sume a stronger logarithmic Sobolev inequality instead of a Poincaré inequality), but the common property is that some growth condition onV is assumed and in particular the measuree^−Vd xis bounded. Here we consider the caseV ≡0, which is obviously a different regime. By replacing the Poincaré inequality by Nash’s in- equality or using direct estimates in Fourier variables, we adapt the L² hypoco- ercivity methods and prove that an appropriate norm of the solution decays at a rate which is the rate of the heat equation. This observation is compatible with diffusion limits, which have been a source of inspiration for building Lyapunov functionals and establishing the L²hypocoercivity method of [11]. Before stating any result, we need some notation to implement thefactorizationmethod of [16]

and obtain estimates in large functional spaces.

Let us consider the measures

(4) dγk:=γk(v)d v where γk(v)=¡

1+ |v|²¢k/2

and k>d,

such that 1/γk∈L¹(R^d). The conditionk∈(d,∞] then covers the case of weights with a growth of the order of|v|^k, whenkis finite, and we denotek= ∞the case when the weightγ∞=M⁻¹grows at least exponentially fast.

Theorem 1. — Assume (H1)–(H4), x ∈R^d, and k ∈(d,∞]. Then there exists a constant C >0such that solutions f of (1) with initial datum f0∈L²(d x dγk)∩ L²(dγk; L¹(d x))satisfy, for all t≥0,

°

°f(t,·,·)°

°

2

L²(^{d x d}γk)≤C

°

°f0°

°

2

L²(^{d x d}γk)+°

°f0°

°

2

L²(^dγk; L¹(d x)) (1+t)^d/2 .

For the heat equation improved decay rates can be shown by Fourier tech- niques, if the modes with slowest decay are eliminated from the initial data. The following two results are in this spirit.

Theorem 2. — Let the assumptions of Theorem1hold, and let Ï

R^d×R^df0d x d v=0 .

Then there exists C>0such that solutions f of (1)with initial datum f0satisfy, for all t≥0,

°

°f(t,·,·)°

°

2

L²(d x dγk)≤C

°

°f0

°

°

2

L²(dγk+2; L¹(d x))+°

°f0

°

°

2

L²(dγk; L¹(|x|d x))+°

°f0

°

°

2 L²(d x dγk)

(1+t)^d/2+1 ,

with k∈(d,∞).

The case of Theorem2, but with k = ∞, is covered in Theorem 3under the stronger assumption that M is a Gaussian. For the formulation of a result cor- responding to the cancellation of higher order moments, we introduce the set R<sub>`</sub>[X,V] of polynomials of order at most`in the variablesX,V ∈R^d(the sum of the degrees inXand inV is at most`). We also need that the kernel of the collision operator is spanned by a Gaussian function in order to keep polynomial spaces in- variant. This means that for anyP ∈R`[X,V], one has (

L

−

T

) (P M)∈R`[X,V]M. Since the transport operator mixes both variables xand v, one needs moments with respect to bothxandvvariables.

Theorem 3. — In Case (a), let M be the normalized Gaussian (2). In Case (b), we assume thatσ ≡1. Let k ∈(d,∞], `∈Nand assume that the initial datum

f0∈L¹(R^d×R^d)is such that (5)

Ï

R^d×R^df₀(x,v)P(x,v)d x d v=0

for all P ∈R`[X,V]. Then there exists a constant c_k >0such that any solution f of (1)with initial datum f0satisfies, for all t≥0,

°

°f(t,·,·)°

°

2

L²(d x dγk)≤c_k

°

°f0

°

°

2

L²(dγk+2; L¹(d x))+°

°f0

°

°

2

L²(dγk; L¹(|x|d x))+°

°f0

°

°

2 L²(d x dγk)

(1+t)^d/2+1+` .

Theoutline of this papergoes as follows. In Section2, we slightly strengthen theabstract hypocoercivityresult of [11] by allowing complex Hilbert spaces and by providing explicit formulas for the coefficients in the decay rate (Proposition4). In Corollary5, this result is applied for fixedξto the Fourier transformed problem (3), where integrals are computed with respect to the measure dγ∞ in the velocity variablev. Since the frequencyξcan be considered as a parameter, we shall speak of amode-by-mode hypocoercivityresult. It provides exponential decay, however with a rate deteriorating asξ→0.

In Section 3, we state a special case (Proposition 6) of the factorization re- sult of [16] with explicit constants which corresponds to anenlargement of the space, and also a shrinking result (Proposition 7) which will be useful in Sec- tion6.2. By the enlargement result, the estimate corresponding to the exponential weightγ∞is extended in Corollary8to larger spaces corresponding to the alge- braic weightsγk withk∈(d,∞). As a straightforward consequence, in Section4, we recover anexponential convergence ratein the case of the flat torusT^d(Corol- lary9), and then give a first proof of thealgebraic decay rateof Theorem1in the whole space without confinement.

In Section5, an hypocoercivity method, where the Poincaré inequality, or the so-calledmacroscopic coercivitycondition, is replaced by theNash inequality, pro- vides an alternative proof of Theorem1. Such a direct approach is also applica- ble to problems with non-constant coefficients like scattering operators withx- dependent scattering ratesσ, or Fokker-Planck operators withx-dependent diffu- sion constants like∇v·¡

D(x)M∇v(M⁻¹f)¢ .

Theimproved algebraic decay ratesof Theorem2and Theorem3are obtained by direct Fourier estimates in Section6. As we shall see in the Appendix A, the rates of Theorem 1are optimal: the decay rate is the rate of the heat equation onR^d. Our method is consistent with thediffusion limit and provides estimates which are asymptotically uniform in this regime: seeAppendix B. We also check that the results of Theorem2and Theorem3are uniform in the diffusive limit in Appendix B.

We conclude this introduction by a briefreview of the literature:On the whole Euclidean space, we refer to [31] for recent lecture notes on available techniques for capturing the large time asymptotics of the heat equation. Some of our results make a clear link with the heat flow seen as the diffusion limit of the kinetic equa- tion. We also refer to [21] for recent results on the diffusion limit, or overdamped limit (seeAppendix B).

The mode-by-mode analysis is an extension of the hypocoercivity theory of [11], which has been inspired by [18], but is also close to the Kawashima compensating function method: see [24] and [15, Chapter 3, Section 3.9]. We also refer to [12]

where the Kawashima approach is applied to a particular case of the scattering model (b).

The wordhypocoercivitywas coined by T. Gallay and widely disseminated in the context of kinetic theory by C. Villani. In [28,33,34], the method deals with large time properties of the solutions by considering a H¹-norm (inxandvvariables) and taking into account cross-terms. This is very well explained in [33, Section 3], but was already present in earlier works like [19]. Hypocoercivity theory is inspired by and related to the earlierhypoellipticity theory. The latter has a long history in the context of the kinetic Fokker-Planck equation. One can refer for instance to [13,19] and much earlier to Hörmander’s theory [20]. The seed for such an approach can even be traced back to Kolmogorov’s computation of Green’s kernel for the kinetic Fokker-Planck equation in [25], which has been reconsidered in [22]

and successfully applied, for instance, to the study of the Vlasov-Poisson-Fokker- Planck system in [32,6].

Linear Boltzmann equations and BGK (Bhatnagar-Gross-Krook, see [5]) mod- els also have a long history: we refer to [9, 8] for key mathematical properties, and to [28,18] for first hypocoercivity results. In this paper we will mostly rely on [10,11]. However, among more recent contributions, one has to quote [17,1,7]

and also an approach based on the Fisher information which has recently been implemented in [14,27].

With theexponential weightγ∞=M⁻¹, Corollary9can be obtained directly by the method of [11]. In this paper we also obtain a result for weights with poly- nomial growth in the velocity variable based on [16]. For completeness, let us mention that recently the exponential growth issue was overcome for the Fokker- Planck case in [23,26] by a different method. The improved decay rates estab- lished in Theorem2and in Theorem3generalize to kinetic models similar results known for the heat equation, see for instance [26, Remark 3.2 (7)] or [3].

2. Mode-by-mode hypocoercivity Let us consider the evolution equation

(6) d F

d t +

T

F=

L

F,

where

T

and

L

are respectively a generaltransport operatorand a generallinear collision operator. We shall use the abstract approach of [11]. Although the exten- sion of the method to Hilbert spaces over complex numbers is rather straightfor- ward, we carry it out here for completeness. For details on the Cauchy problem or, e.g., on the domains of the operators, we refer to [11]. Notice that we do not ask that

L

is a Hermitian operator but simply assume that

L

^∗

A

=0.

Proposition 4. — Let

L

and

T

be closed unbounded linear operators on the com- plex Hilbert space(H,〈·,·〉)with dense domainsD(L)andD(T). Assume that

T

is anti-Hermitian. LetΠbe the orthogonal projection onto the null space of

L

and define

A

:=¡

1+(

T

_Π)^∗

T

_Π^¢−1(

T

_Π)^∗

where^∗denotes the adjoint with respect to〈·,·〉. We assume that

L

^∗

A

₌0and that there are positive constantsλm,λM, and C_M exist, such that, for any F ∈H, the following properties hold:

B

microscopic coercivity:

(A1) − 〈

L

F,F〉 ≥λmk(1−Π)Fk², ∀F∈D(L) ,

B

macroscopic coercivity:

(A2) k

T

_ΠFk²≥λMkΠFk², ∀F∈D(T) ,

B

parabolic macroscopic dynamics:

(A3) Π

T

_ΠF=0, ∀F∈D(T) ,

B

bounded auxiliary operators:

(A4) k

AT

(1−Π)Fk + k

AL

Fk ≤CMk(1−Π)Fk, ∀F∈D(L)∩D(T) . Then L−T generates a C₀-semigroup and for any t≥0, we have (7) °

°

°e^(L−T)t°

°

°

2

≤3e^−λt where λ= λM

3 (1+λM)min (

1,λm, λmλM

(1+λM)C²_M )

. Proof. — For someδ>0 to be determined later, the Lyapunov functional

H

[F] :=¹₂kFk²+δRe〈

A

F,F〉

is such that _{d t}^d

H

[F]= −

D

[F] ifFsolves (6), with

D

[F] := −〈

L

F,F〉 +δ〈

AT

_ΠF,F〉 +δRe〈

AT

(1−Π)F,F〉 −δRe〈

TA

F,F〉 −δRe〈

AL

F,F〉. Note that we have used the fact that Re〈

A

F,

L

F〉 =0 because of the assumption

L

^∗

A

₌0, and also that〈

AT

_ΠF,F〉is real because

AT

_Πis self-adjoint by construc- tion. Since the Hermitian operator

AT

_Πcan be interpreted as the application of the mapz7→(1+z)⁻¹z to (

T

_Π)^∗

T

_Π and as a consequence of the spectral theo- rem [30, Theorem VII.2, p. 225], the conditions (A1) and (A2) imply that

− 〈

L

F,F〉 +δ〈

AT

_ΠF,F〉 ≥λmk(1−Π)Fk²+ δλM

1+λM kΠFk². As in [11, Lemma 1], ifG=

A

F,i.e.,G+(

T

_Π)^∗

T

_ΠG=(

T

_Π)^∗F, one has

k

A

Fk²+ k

TA

Fk²= 〈G,G+(

T

_Π)^∗

T

_ΠG〉 = 〈G, (

T

_Π)^∗F〉 = 〈

TA

F, (

1

_−Π)F〉

where we have used A =ΠA andΠTΠ=0. Using|〈

TA

F, (

1

_{−Π)F〉| ≤ k}

TA

Fk²+

1

4k(

1

−Π)Fk², one gets

(8) k

A

Fk²≤1

4k(

1

_−Π)Fk²,

which implies that|Re〈

A

F,F〉| ≤ kAFkkFk ≤¹₂kFk²and provides us with the norm equivalence of

H

[F] andkFk²,

(9) 1

2(1−δ)kFk²≤

H

[F]≤1

2(1+δ)kFk². WithX:= k(1−Π)FkandY := kΠFk, it follows from (A4) that

D

[F]≥(λm−δ)X²+ δλM

1+λM

Y²−δCMX Y. The choiceδ=¹₂minn

1,λm,_(1+λ^λmλM

M)C²_M

o

implies that

D

[F]≥λm

4 X²+ δλM

2 (1+λM)Y²≥1 4min

½

λm,2δλM

1+λM

¾

kFk²≥ 2δλM

3 (1+λM)

H

[F] . Withλdefined in (7), usingδ≤1/2 and (1+δ)/(1−δ)≤3, we get

kF(t)k²≤ 2

1−δ

H

[F](t)≤1+δ

1−δe^−λtkF(0)k²≤3e^−λtkF(0)k². For any fixedξ∈R^d, let us apply Proposition4to (3) withF=fˆand H =L²¡

dγ∞

¢, kFk²= Z

R^d|F|²dγ∞, ΠF=M Z

R^dF d v=MρF,

T

F=i(v·ξ)F. Here we are in a mode-by-mode framework in which the transport operator

T

is a simple multiplication operator.

Corollary 5. — Assume(H1)–(H4), and takeξ∈R^d. If f is a solution ofˆ (3)such thatfˆ₀(ξ,·)∈L²(dγ∞), then for any t≥0, we have

°

°fˆ(t,ξ,·)°

°

2

L²(^dγ∞)≤3e^−µξt°

°fˆ0(ξ,·)°

°

2 L²(^dγ∞) , where

(10) µξ:= Λ|ξ|²

1+ |ξ|² and Λ=1 3min©

1,Θª min

½

1, λmΘ² K+Θκ²

¾ , with

(11) Θ:= Z

R^d(v·

e

)²M(v)d v, K:= Z

R^d(v·

e

)⁴M(v)d v, θ:= 4 d Z

R^d

¯

¯¯∇v

pM¯

¯

¯

2d v, for an arbitrary

e

_∈S^d−1, and withκ=p

θin Case (a) andκ=2σp

Θin Case (b).

Proof. — We check that the assumptions of Proposition4are satisfied withF=fˆ. The property

L

^∗

A

₌0 is a consequence of the mass conservationR

R^d

L

f d v =0 becauseΠ

A

₌

A

. Assumption (H4) implies (A1). Concerning the macroscopic coercivity (A2), since

T

_ΠF=i(v·ξ)ρFM, one has

k

T

_ΠFk²_{= |ρF|}² Z

R^d|v·ξ|²M(v)d v =Θ|ξ|²|ρF|²=Θ|ξ|²kΠFk²,

and thus (A2) holds withλM=Θ|ξ|². By assumptionM(v) depends only on|v|, so it isunbiased:R

R^dv M(v)d v=0, which means that (A3) holds.

Let us now prove (A4). Since (

T

_Π)^∗F = −Π

T

F= −i(ξ·R

R^dv⁰F(v⁰)d v⁰)M, we obtain that

¡1+(

T

_Π)^∗

T

_Π^¢_ρM= µ

1+ Z

R^d

¡ξ·v⁰¢2

M(v⁰)d v⁰

¶

ρM=¡

1+Θ|ξ|²¢ ρM and the operator

A

, defined in Proposition4, is given mode-by-mode by

A

F=−iξ·R

R^dv⁰F(v⁰)d v⁰ 1+Θ|ξ|² M. As a consequence,

A

satisfies the estimate

k

A

Fk = k

A

(

1

_−Π)Fk ≤ 1 1+Θ|ξ|²

Z

R^d

|(

1

_−Π)F|

pM |v·ξ|p M d v

≤k(

1

−Π)Fk 1+Θ|ξ|²

µZ

R^d(v·ξ)²M d v

¶1/2

=

pΘ|ξ|

1+Θ|ξ|²k(

1

_−Π)Fk.

In Case (b) the collision operator

L

is obviously bounded:

k

L

_{Fk ≤}2σk(

1

_−Π)Fk and, as a consequence,

k

AL

Fk ≤2σp Θ|ξ|

1+Θ|ξ|² k(

1

_−Π)Fk.

We also notice that

L

^∗

A

₌0 according to (H3). For estimating

AL

in Case (a), we

note that Z

R^dv

L

F d v=2 Z

R^d∇v

pM F pMd v and obtain as above that

k

AL

Fk ≤ 2 1+Θ|ξ|²

Z

R^d

|(

1

_−Π)F| pM

¯

¯¯ξ· ∇v

pM¯

¯

¯d v≤

pθ|ξ|

1+Θ|ξ|²k(

1

_−Π)Fk. For both cases we finally obtain

k

AL

Fk ≤ κ|ξ|

1+Θ|ξ|²k(

1

_−Π)Fk. Similarly we can estimate

AT

(

1

−Π)F=

R

Rd(^v0·ξ)²(1−Π)F(v⁰)d v⁰

1+Θ|ξ|² Mby

k

AT

(

1

−Π)Fk =

¯

¯

¯ R

R^d¡ v⁰·ξ¢2

(1−Π)F(v⁰)d v⁰¯

¯

¯ 1+Θ|ξ|²

≤

³R

R^d¡ v⁰·ξ¢4

M(v⁰)d v⁰´1/2

1+Θ|ξ|² k(

1

−Π)Fk =

pK|ξ|²

1+Θ|ξ|²k(

1

−Π)Fk, meaning that we have proven (A4) withCM =^κ|ξ|+

pK|ξ|² 1+Θ|ξ|² . With the elementary estimates

Θ|ξ|²

1+Θ|ξ|²≥min©

1,Θª |ξ|²

1+ |ξ|² and λM

(1+λM)C_M² =Θ¡

1+Θ|ξ|²¢

¡κ+p

K|ξ|¢2≥ Θ² K+Θκ², the proof is completed using (7).

3. Enlarging and shrinking spaces by factorization

Square integrability against the inverse of thelocal equilibrium Mis a rather re- strictive assumption on the initial datum. In this section it will be relaxed with the help of the abstractfactorization methodof [16] in a simple case (factorization of order 1). Here we state the result and sketch a proof in a special case, for the con- venience of the reader. We shall then give a result based on similar computations in the opposite direction: how to establish a rate in a stronger norm, which corre- spond to ashrinkingof the functional space. We will conclude with an application to the problem studied in Corollary5. Let us start byenlargingthe space.

Proposition 6. — LetB1,B2be Banach spaces and letB2be continuously imbed- ded inB1,i.e.,k · k1≤c1k · k2. Let

B

_and

A

₊

B

be the generators of the strongly continuous semigroups e^Btand e^(A+B)tonB1. Assume that there are positive con- stants c2, c3, c4,λ1andλ2such that, for all t≥0,

°

°

°e^(A+B)t°

°

°2→2≤c2e^−λ2t, °

°

°e^Bt°

°

°1→1≤c3e^−λ1t, k

A

_k1→2≤c4,

wherek · ki→jdenotes the operator norm for linear mappings fromBi toBj. Then there exists a positive constant C=C(c1,c2,c3,c4)such that, for all t≥0,

°

°

°e^(A+B)t°

°

°₁

→1≤ ( C¡

1+ |λ1−λ2|⁻¹¢

e^{−min{λ1,λ2}t} forλ16=λ2, C(1+t)e^−λ1t forλ1=λ2. Proof. — Integrating the identity _{d s}^d ¡

e^(A+B)se^B(t−s)¢

=e^(A+B)s

A

e^B(t−s)with re- spect tos∈[0,t] gives

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

0

e^(A+B)s

A

e^B(t−s)d s. The proof is completed by the straightforward computation

ke^(A+B)tk1→1≤c₃e^−λ1t+c₁ Z t

0 ke^(A+B)s

A

e^B(t−s)k1→2d s

≤c3e^−λ1t+c1c2c3c4e^−λ1t Z t

0

e^(λ1−λ2)sd s.

The second statement of this section is devoted to a result on theshrinkingof the functional space. It is based on a computation which is similar to the one of the proof of Proposition6.

Proposition 7. — LetB1,B2be Banach spaces and letB2be continuously imbed- ded inB1,i.e.,k · k1≤c1k · k2. Let

B

and

A

₊

B

be the generators of the strongly continuous semigroups e^Btand e^(A+B)tonB1. Assume that there are positive con- stants c2, c3, c4,λ1andλ2such that, for all t≥0,

°

°

°e^(A+B)t

°

°

°₁

→1≤c2e^−λ1t,

°

°

°e^Bt

°

°

°₂

→2≤c3e^−λ2t, k

A

_k1→2≤c4,

wherek · ki→jdenotes the operator norm for linear mappings fromBi toBj. Then there exists a positive constant C=C(c₁,c₂,c₃,c₄)such that, for all t≥0,

°

°

°e^(A+B)t°

°

°2→2≤ ( C¡

1+ |λ2−λ1|⁻¹¢

e^{−min{λ2,λ1}t} forλ26=λ1, C(1+t)e^−λ1t forλ1=λ2.

Proof. — Integrating the identity _{d s}^d ¡

e^B(t−s)e^(A+B)s¢

=e^B(t−s)

A

e^(A+B)s with re- spect tos∈[0,t] gives

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

0

e^B(t−s)

A

e^(A+B)sd s. The proof is completed by the straightforward computation

ke^(A+B)tk_2→2≤c3e^−λ2t+ Z t

0 ke^B(t−s)

A

e^(A+B)sk_2→2d s

≤c₃e^−λ2t+c₁ Z t

0 ke^B(t−s)

A

e^(A+B)sk1→2d s

≤c3e^−λ2t+c1

Z t

0 ke^B(t−s)k_2→2k

A

_k1→2ke^(A+B)sk_1→1d s

≤c3e^−λ2t+c1c2c3c4e^−λ2t Z t

0

e^(λ2−λ1)sd s.

We will use Proposition7in Section6.2. Coming back to the problem studied in Corollary5, Proposition6applies to (3) with the spacesB1=L²(dγk),k∈(d,∞), andB2=L²(dγ∞) corresponding to the weights defined by (4). The exponential growth ofγ∞guarantees thatB2is continuously imbedded inB1.

Corollary 8. — Assume (H1)–(H4), k ∈(d,∞], and ξ∈R^d. Then there exists a constant C >0, such that solutions f ofˆ (3)with initial datum fˆ0(ξ,·)∈L²(dγk) satisfy, withµ_ξgiven by(10),

°

°fˆ(t,ξ,·)°

°

2

L²(dγk)≤C e^−µξt°

°fˆ₀(ξ,·)°

°

2

L²(dγk) ∀t≥0 .

Proof. — In Case (a), let us define

A

_and

B

_by

A

F=NχRFand

B

F= −i(v·ξ)F+

L

F−

A

F, whereN andRare two positive constants,χis a smooth function such that1B₁≤χ≤1B₂, andχR :=χ(·/R). HereBr is the centered ball of radiusr. It has been established in [26, Lemma 3.8] that ifk>d, then the inequality

Z

R^d(

L

₋

A

)(F)F dγk≤ −λ1

Z

R^dF²dγk

holds for someλ1>0. Moreover,λ1 can be chosen arbitrarily large forR andN large enough. The boundedness of

A

:B1→B2follows from the compactness of the support ofχand Proposition6applies withλ2=µξ/2≤1/4, whereµξis given by (10).

In Case (b), we consider

A

and

B

such that

A

F(v) = M(v)

Z

R^dσ(v,v⁰)F(v⁰)d v⁰,

B

F(v) = −

·

i(v·ξ)+ Z

R^dσ(v,v⁰)M(v⁰)d v⁰

¸ F(v) . The boundedness of

A

_:B1→B2follows from (H2) and

k

A

FkL²(dγ∞)≤σkFkL¹(d v)≤σ µZ

R^dγ⁻¹_k d v

¶1/2

kFkL²(dγk). Proposition6applies withλ2=^µ₂^ξ≤¹₄andλ1=1 becauseR

R^dσ(v,v⁰)M(v⁰)d v⁰≥1.

4. Asymptotic behavior based on mode-by-mode estimates

In this section we consider (1) and use the estimates of Corollary5with weight γ∞=1/Mand Corollary8for weights withO(|v|^k) growth to get decay rates with respect tot. We shall consider two cases for the spatial variablex. In Section4.1, we assume that x ∈T^d, where T^d is the flatd-dimensional torus (represented by [0, 2π)^dwith periodic boundary conditions) and prove an exponential conver- gence rate. In Section4.2, we assume thatx∈R^d and establish algebraic decay rates.

4.1. Exponential convergence to equilibrium inT^d. — In the periodic casex∈ T^dthere is a unique non-zero normalized equilibrium given by

f_∞(x,v)=ρ∞M(v) with ρ∞= 1

|T^d| Ï

T^d×R^df₀d x d v.

Corollary 9. — Assume (H1)–(H4) and k ∈(d,∞]. Then there exists a constant C>0, such that the solution f of (1)onT^d×R^dwith initial datum f₀∈L²(d x dγk) satisfies, withΛgiven by(10),

°

°f(t,·,·)−f_∞°

°_L2(d x dγk)≤C°

°f₀−f_∞°

°_L2(d x dγk)e^−Λ4t ∀t≥0 .

Proof. — We represent the flat torusT^d by [0, 2π)^d with periodic boundary con- ditions, and the Fourier variable is denotedξ∈Z^d. Forξ=0, the microscopic coercivity (see Section2) implies

°

°fˆ(t, 0,·)−fˆ_∞(0,·)°

°_L2(dγ∞)≤°

°fˆ₀(0,·)−fˆ_∞(0,·)°

°_L2(dγ∞)e^−t.

For all other modes, ˆf_∞(ξ,·)=0 for anyξ6=0 (that is, for anyξsuch that|ξ| ≥1).

We can use Corollary5withµξ≥Λ/2, with the notations of (10). An application of Parseval’s identity then proves the result fork= ∞, andC=p

3. Ifk is finite, the result with the weightγkfollows from Corollary8.

Note that the latter result can also alternatively be proved by directly applying Proposition4to (1), as in [11].

4.2. Algebraic decay rates inR^d. — With the result of Corollary5and Corollary8 we obtain a first proof of Theorem1as follows. LetC >0 be a generic constant which is going to change from line to line. Plancherel’s formula implies

°

°f(t,·,·)°

°

2

L²(d x dγk)≤C Z

R^d

µZ

R^de^−µξt¯

¯fˆ₀¯

¯

2dξ

¶ dγk. We know thatR

|ξ|≤1e^−µξtdξ≤R

R^de^−Λ2|ξ|2tdξ=¡_2π

Λt

¢d/2

and thus, for allv∈R^d, Z

|ξ|≤1e^−µξt¯

¯fˆ0

¯

¯

2dξ≤C°

°f0(·,v)°

°

2 L¹(d x)

Z

R^de^−Λ2|ξ|2tdξ≤C°

°f0(·,v)°

°

2

L¹(d x)t^−d2. Using the fact thatµξ≥Λ/2 when|ξ| ≥1 and Plancherel’s formula, we know that, for allv∈R^d,

Z

|ξ|>1e^−µξt¯

¯fˆ0

¯

¯

2dξ≤C e^−Λ2t°

°f0(·,v)°

°

2 L²(d x), which completes a first proof of Theorem1.

5. Hypocoercivity and the Nash inequality

In view of the proof of Theorem1in Section4.2and of the rate, it is natural to wonder if the hypocoercivity can be controlled by the use of Nash’s inequality. Here we temporarily abandon the Fourier variableξand consider the direct variable x ∈R^d: throughout this section, thetransport operator on the position space is defined as

T

f =v· ∇xf.

We rely on the abstract setting of Section2, applied to (1) with the scalar product

〈·,·〉on L²(d x dγ∞) and the induced normk · k. Notice that this norm includes the xvariable, which was not the case in the mode-by-mode analysis of Section2. It is then easy to check that (

T

_Π)f =M

T

_ρf =v·∇xρfM, (

T

_Π)^∗f = −∇x·¡R

R^dv f d v¢ M and (

T

_Π)^∗(

T

_Π)f = −Θ∆xρfMso that

g=

A

f =¡

1+(

T

_Π)^∗

T

_Π^¢−1(

T

_Π)^∗f ⇐⇒ g=u M whereu−Θ∆u= −∇x·¡R

R^dv f d v¢

. SinceMis unbiased,

A

f =

A

(

1

−Π)f. For some δ>0 to be chosen later, we redefine the entropy by

H

[f] :=¹₂kfk²+δ〈

A

f,f〉.

Proof of Theorem1. — Iff solves (1), the time derivative of

H

[f(t,·,·)] is given by

(12) d

d t

H

[f]= −

D

[f] where, as in the proof of Proposition4,

D

[f] := −〈

L

f,f〉+δ〈

AT

_Πf,f〉+δRe〈

AT

(1−Π)f,f〉−δRe〈

TA

f,f〉−δRe〈

AL

f,f〉. Here we use the fact that〈

A

f,

L

f〉 =0. The first term in

D

[f] satisfies the micro- scopic coercivity condition

− 〈

L

f,f〉 ≥λmk(

1

_−Π)fk².

The second term in (12) is computed as follows. Solvingg=

AT

_Πf is equivalent to solving (1+(

T

_Π)^∗

T

_Π)g=(

T

_Π)^∗

T

_Πf,i.e.,

(13) v_f−Θ∆xv_f = −Θ∆xρf,

whereg=v_f M. Hence

〈

AT

_Πf,f〉 = Z

R^dv_fρfd x.

A direct application of the hypocoercivity approach of [11] to the whole space problem fails by lack of amacroscopic coercivitycondition. Although the second term in (12) is not coercive, we observe that the last three terms in (12) can still be dominated by the first two forδ>0, small enough, as follows.

1) As in [11], we use the adjoint operators to compute

〈

AT

(

1

−Π)f,f〉 = −〈(

1

−Π)f,

TA

^∗f〉.

We observe that

A

^∗f =

T

_Π^¡1+(

T

_Π)^∗

T

_Π^¢−1f =

T

^¡1+(

T

_Π)^∗

T

_Π^¢−1_Πf =M

T

u_f =v M· ∇xu_f whereuf is the solution in H¹(d x) of

(14) uf −Θ∆xuf =ρf.

WithK defined by (11), we obtain that

°

°

TA

^∗f°

°

2≤K°

°∇²xu_f°

°

2

L²(d x)=K °

°∆xu_f°

°

2 L²(d x).

On the other hand, we observe thatv_f = −Θ∆u_f solves (13). Hence by multiply- ing (14) byvf= −Θ∆uf and integrating by parts, we know that

(15) Θ°

°∇xuf

°

°

2

L²(d x)+Θ²°

°∆xuf

°

°

2 L²(d x)=

Z

R^dvfρfd x= 〈

AT

_Πf,f〉.

Notice that a central feature of our method is the fact that quantities of interest involving the operator

A

can be computed by solving an elliptic equation (for in- stance (13) in case of

AT

_Πf or (14) in case of

A

^∗f). Altogether we obtain that

|〈

AT

(

1

_−Π)f,f〉| ≤ k(

1

−Π)fk k

TA

^∗fk ≤ pK

Θ k(

1

−Π)fk 〈

AT

_Πf,f〉^1/2. 2) By (8), we have

¯

¯〈

TA

f,f〉¯

¯=¯

¯〈

TA

(

1

_−Π)f, (

1

_−Π)f〉¯

¯≤ k(

1

_−Π)fk².

3) It remains to estimate the last term on the right hand side of (12). Let us consider the solutionuf of (14). If we multiply (13) byuf and integrate, we observe that

Θ°

°∇xuf

°

°

2 L²(d x)=

Z

R^dufvfd x≤ Z

R^dufvfd x+ Z

R^d|vf|²d x= Z

R^dvfρfd x becausev_f = −Θ∆u_f, so that

k

A

^∗fk²=Θ°

°∇xu_f°

°

2

L²(d x)≤ 〈

AT

_Πf,f〉. In Case (a), we compute

〈

AL

f,f〉 = 〈

L

(

1

_−Π)f,

A

^∗f〉 = Ï

R^d×R^d∇xuf·∇vM

M (

1

_−Π)f d x d v. It follows from the Cauchy-Schwarz inequality that

Z

R^d|∇vM| |(

1

_−Π)f|dγ∞≤ k∇vMkL²(^dγ∞)k(

1

_−Π)fkL²(^dγ∞)

=p

dθk(

1

−Π)fkL²(^dγ∞) and

|〈

AL

f,f〉| ≤°

°∇xu_f°

°_{L2(d x)} µZ

R^d

µ1 d Z

R^d|∇vM| |(

1

_−Π)f|dγ

¶2

d x

¶

1 2

. Altogether, we obtain that

|〈

AL

f,f〉| ≤ sθ

Θk(

1

_−Π)fk 〈

AT

_Πf,f〉¹². In Case (b), we use (H2) to get that

|〈

AL

f,f〉| ≤ k

L

fk k

A

^∗fk ≤2σk(

1

_−Π)fk k

A

^∗fk ≤2σk(

1

_−Π)fk 〈

AT

_Πf,f〉¹². In both cases, (a) and (b), the estimate can be written as

|〈

AL

f,f〉| ≤2σk(

1

_−Π)fk 〈

AT

_Πf,f〉¹² with the convention thatσ=¹₂p

θ/Θin Case (a).

Summarizing, we know that

−d

d t

H

[f]≥(λm−δ)X²+δY²+2δ

b

X Y

withX := k(

1

_−Π)fk,Y := 〈

AT

_Πf,f〉^1/2 and

b

:= ₂^K_Θ+2σ. The largest

a

>0 such that

(λm−δ)X²+δY²+2δ

b

X Y ≥

a

^¡X²+2Y²¢