A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium

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A note on hypocoercivity for kinetic equations with

heavy-tailed equilibrium

Nathalie Ayi, Maxime Herda, Hélène Hivert, Isabelle Tristani

To cite this version:

Nathalie Ayi, Maxime Herda, Hélène Hivert, Isabelle Tristani. A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium. 2019. �hal-02389146�

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A NOTE ON HYPOCOERCIVITY FOR KINETIC EQUATIONS WITH HEAVY-TAILED EQUILIBRIUM

NATHALIE AYI, MAXIME HERDA, H´EL`ENE HIVERT, AND ISABELLE TRISTANI

Abstract. In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic L´evy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local L´evy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.

Keywords. Hypocoercivity; linear kinetic equations; Fokker-Planck operator; fractional diffu-sion; heavy-tailed distribution.

2010 Mathematics Subject Classification. 82C40; 35K65; 35Q84; 60G22.

Contents

1. Introduction 1

2. The L´evy-Fokker-Planck operator as bilinear form 3

3. Coercivity results for the L´evy-Fokker-Planck operator 4

4. An interpolation inequality 5

5. Proof of Theorem 1.1 6

6. The case of the heavy-tailed BGK equation 8

Acknowledgements 9

References 9

1. Introduction

We consider a distribution functionf ”fpt, x, vqwhich depends on timetě0, positionxPTd

and velocity vPRd and satisfies the fractional kinetic Fokker-Planck equation (1) Btf`v¨∇xf “ ∇v¨ pvfq ´ p´∆vqα{2f .

Here we assumeαP p0,2qand the fractional Laplacian´p´∆vqα{2 is such that for any Schwartz

function g:RdÑ R, one hasF_pp´_∆_v_qα{2_g_qp_ξ_{q “ |}_ξ_|α_F_p_g_qp_ξ_q _where_F_p¨q _{denotes the Fourier}

transform. There are many equivalent definitions of the fractional Laplacian (see [11]). Among them we shall use (here, P.V.stands for the principal value)

(2) p´∆vqα{2gpvq “ Cd,αP.V. ż Rd gpvq ´gpwq |v´w|d`α dw , 1

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where the constant Cd,α is given by Cd,α “2αΓpd`₂αq{pπd{2|Γp´α₂q|q where Γp¨q is the Gamma

function. In the following we drop the principal value in the notations. We denote the L´evy-Fokker-Planck operator appearing on the right-hand side of (1) by

Lαg “ ∇v¨ pv gq ´ p´∆vqα{2g .

By passing to Fourier variables one has F_pLαgqpξq “ ´ξ ¨∇ξˆgpξq ´ |ξ|αgˆpξq, where ˆg “ Fpgq.

From this formula, one sees that the function µαpvq “ Zd,α´1F´

1´_e´|ξ|α_{_α¯

with Zd,α chosen such that ş

µα “1 is a probability distribution such thatLαµα “0. Observe

that away from the origin, the Fourier transform of µα is smooth and rapidly decaying at

infinity. The singularity at ξ “ 0 behaves like |ξ|α _{at principal order which yields that}_µ αpvq

should decay as |v|´α´d _when _|_v_{| Ñ 8}_{. Actually, one has the following more precise bounds}

(see [1] and references therein). There exist some positive constants C1 “ C1pα, dq ą 0 and C2“C2pα, dq ą0 such that for allvPRd one has

(3) C₁´1 ď p|v|d`α_`₁_q_µ

αpvq ď C1, and

(4) C₂´1|v| ď p|v|2`d`α`1q|∇_vµαpvq| ď C2|v|.

In the following, given some measurable non-negative function ν ”νpvq we denote by L2_vpνq

and L2_x,vpνq the spaces of measurable functions g of respectively the v and the px, vq variables such that |g|2ν is integrable. We endow these spaces with their canonical scalar product and norm. We also introduce the corresponding Sobolev space H_x,v1 pνq associated with the norm

}g}2_H1_x,vp_ν_q “ }g}2_L2_x,vp_ν_q` }∇xg}2L2_x,vp_ν_q` }∇vg}2L2

x,vpνq.

Finally given an integrable function g, we denotexgy “ť_Td_ˆ_Rdgpx, vqdvdx the global mass of

g. The main result of this paper is the following.

Theorem 1.1. Let f be a solution of the kinetic L´evy-Fokker-Planck equation (1) with initial data fin

PH_x,v1 pµ´_α1q. Then, for all tě0 one has

}fptq ´@finDµα}_H1_x,vp_µ´1 α q ď C}f in ´@finDµα}_H_x,vp1 _µ´1 α qe ´λt

for some constant C ě1 and λą0 depending only on d and α.

Let us mention that these results have been obtained as a preliminary step towards the con-ception and analysis of numerical schemes preserving the long-time behavior of these equations. This topic is an ongoing work [2] in the spirit of what has previously been done in [7, 3] in the case of the classical Fokker-Planck equation. The compatibility of our schemes with anomalous diffusion limit will also be investigated (see [6] for more details).

Before going into the analysis of our problem, let us recall that results on large time behavior of solutions to the homogeneous version of (1), namely Btfpt, vq “Lαfpt, vq, have been obtained

in [8] in spaces of type L2_vpµ´_α1q (among others) and later in [12] in larger Lebesgue spaces. Notice that the presence of the transport operator in our equation (1) makes the analysis more intricate and requires the use of hypocoercivity techniques. In the present note, we useH1 type hypocercivity as presented in [13] or [9] for example. Note also that fractional hypocoercivity

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HYPOCOERCIVITY WITH HEAVY-TAILED EQUILIBRIUM 3 has already been studied recently in [4]. However, the framework is quite different from ours since the phase space isRdˆRd and a L2-hypocoercivity approach is developed.

In the same spirit of our work, let us also mention the paper [10] in which some hypoelliptic estimates are obtained on the non homogeneous fractional Kolmogorov equation (there is no drift term in the studied equation). The method of proof is quite close (based on the use of weighted Lyapunov functional) but the final goal is different in the latter since the main concern is about regularization properties of the equation and not convergence towards the equilibrium. In the present study we focus on a good understanding of the structure of the L´evy-Fokker-Planck operator since we endeavour to carry out our computations as simply as possible in order to adapt our analysis to a discrete framework in [2]. In particular let us point out that we do not need fractional derivatives in our Lyapunov functionals and our proof does not rely on Fourier transform. In this sense our method differs completely from that of [10] and the recent [4] in which a mode by mode analysis is developed.

Outline of the note. From Section 2 to Section 4, we carry out the analysis of the properties of the L´evy-Fokker-Planck operator that will be useful for proving our main result. Then, the proof of Theorem 1.1 is done in Section 5. In the last section we state and prove the equivalent of Theorem 1.1 for the BGK equation with heavy-tailed equilibrium.

Notations. For simplicity, in the subsequent proofs, we denote byC a positive constant depend-ing only on fixed numbers (includdepend-ing dand α) and its value may change from line to line.

2. The L´evy-Fokker-Planck operator as bilinear form

The following quite simple decomposition is actually one of the key elements of our hypocoer-cive analysis carried out in Section 5. Compared to the non-fractional case, we here have a lack of symmetry of our operator in L2_vpµ´_α1q and the following splitting is very helpful to simplify the computations. Moreover, in the non-fractional case, there is a gain of weight in velocity which comes from the particular form of the gradient of the Gaussian equilibrium. Even though we no longer have such a gain in our case, we are still able to close our estimates thanks to the following splitting.

Proposition 2.1. One has the decomposition

´ xLαf, gy_L2

vpµ´α1q “

S_v_p_{f, g}_{q `} A_v_p_{f, g}_q_,

where S_v and A_v are bilinear forms that are respectively symmetric and skew-symmetric and defined by S_v_pf, gq “ Cd,α 2 ĳ Rd_ˆ_Rd “ pf µ´_α1qpvq ´ pf µ´_α1qpwq‰ “pgµ´_α1qpvq ´ pgµ´_α1qpwq‰ |v´w|d`α µαpvqdwdv , and A_v_pf, gq “ Cd,α 2 ĳ Rd_ˆ_Rd pf µ´_α1qpwqpgµ´_α1qpvq ´ pf µ´_α1qpvqpgµ´_α1qpwq |v´w|d`α µαpvqdwdv ` 1 2 ż Rd pf v¨∇_v_pgµ´_α1q ´ g v¨∇_v_pf µ´_α1qqdv , where Cd,α is defined in (2).

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We skip the proof of this proposition since it is based on simple computations using the formula (2), integration by parts and the fact that Lαµα“0.

Observe that a direct consequence of the Cauchy-Schwarz inequality is (5) S_v_pf, gq ď S_v_pf, fq1{2S_v_pg, gq1{2,

forf, gPDpLαq. Moreover, the symmetric formSv is non-negative and Svpf, fq vanishes when

f µ´_α1 is constant. This yields that the nullspace ofLα is exactly given by Rµα. From there the

orthogonal projection Π onto the nullspace of Lα is given by

pΠgqpvq “ ˆż Rd gpwqdw ˙ µαpvq.

3. Coercivity results for the L´evy-Fokker-Planck operator

One has the following coercivity result taken from [8, Theorem 2] and originating from [5].

Lemma 3.1 ([8]). There is a constant CP ”CPpα, dq ą0 such that for all f PDpLαq,

(6) }f´Πf}2_L2

vpµ´α1q ď CP

S_v_pf, fq.

We now show that the dissipation S_v_pf, fq also provides some fractional Sobolev regularity. We introduce the fractional Sobolev space H_vs with s P p0,1q with norm defined by } ¨ }2

Hs v “ } ¨ }2 L2 v ` } ¨ } 2 9 Hs v

where the homogeneous Sobolev norm is given by}g}2

9 Hs v :“ }p´∆qs{2_g_}2 L2 v. One

can prove that there exists a positive constant Crd,s such that

(7) }g}2 9 Hs v “ r Cd,s ĳ Rd_ˆ_Rd |fpvq ´fpwq|2_|_v_´_w_|´pd`2sq_d_w_d_{v .}

Lemma 3.2. There exists CR”CRpα, dq ą0 such that for all f PDpLαq, S_v_p_{f, f}_{q ě} _C´1 R ´ }f µ´_α1{2}2_H₉α{2 v ´ }f µ ´1{2 α }2L2 v ¯ .

Proof. Using thatpa`bq2_ě_a2_{₂_´_b2_{, we have}

S_v_pf, fq ěCd,α 2 ĳ |v´w|ď1 |pµ´_α1fqpvq ´ pµ´_α1fqpwq|2 |v´w|d`α µαpvqdwdvě Cd,α 2 ˆ 1 2I1´I2 ˙ .

The first term is I1 “ ĳ |v´w|ď1 |pµ´α1{2fqpvq ´ pµ´α1{2fqpwq|2 |v´w|d`α dwdv “ Cr_d,´1α 2} µ´_α1{2f}2₉ Hvα{2 ´ ĳ |v´w|ě1 |pµ´α1{2fqpvq ´ pµ´α1{2fqpwq|2 |v´w|d`α dwdv ě Cr_d,´1α 2} µ´_α1{2f}2_H₉α{2 v ´ C}µ´_α1{2f}2_L2 v

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HYPOCOERCIVITY WITH HEAVY-TAILED EQUILIBRIUM 5 where Crd,α₂ is defined in (7) and for the last inequality, we used twice the integrability of |v´

w|´d´α₁

|v´w|ě1, once in v and once inw. The second term is

I2 “ ĳ |v´w|ď1 |µ1α{2pvq ´µ1α{2pwq|2 |v´w|d`α |fpwq| 2_|_µ´1 α pwq|2dwdv .

To treat I2, we use Taylor formula to write

I2“ ĳ |w|ď1 ˇ ˇ ˇş10∇pµ 1{2 α qpv´θwq ¨wdθ ˇ ˇ ˇ2 |w|d`α µ ´1 α pv´wq|fpv´wqµα´1{2pv´wq|2dwdv .

Performing now the changes of variables vÑv´θw and thenθÑ1´θ, we get: I2 ď ĳ |w|ď1 ż1 0 |∇_pµ1α{2qpvq|2 |w|d`α´2 µ ´1 α pv´θwq|fpv´θwqµα´1{2pv´θwq|2dθdwdv .

Notice that, using (3) and since|w| ď1, we haveµ´_α1pv´θwq ďCp1` |v|d`α_q_._{Then, using (4),}

one can prove that |∇_pµ1α{2qpvq|2µ´α1pv´θwq ďC. Consequently, we obtain

I2ďC ĳ |w|ď1 ż1 0 1 |w|d`α´2|fpv´θwqµ ´1{2 α pv´θwq|2dθdwdv

and thus performing a change of variable I2 ďC}f µ´α1{2}2_L2

v.This ends the proof.

Proposition 3.3. There is CF ”CFpα, dq such that for all f PDpLαq,

(8) }pf ´Πfqµ´_α1{2}2

Hvα{2 ď

CFSvpf, fq.

Proof. Let us now summarize the estimates that we have obtained in the two previous lemma. We have S_v_pf, fq ěC_P´1}f´Πf}2 L2 vpµ ´1{2 α q and S_v_pf, fq ě C_R´1`}f µ´_α1{2}2₉ Hvα{2 ´ } f µ´_α1{2}2_L2 v ˘ . Moreover, one can notice thatS_v_p_{f, f}_{q “} S_v_p_f_´_Π_{f, f}_´_Π_f_q_{. As a consequence, an appropriate}

convex combination of the two previous inequalities shows (8).

4. An interpolation inequality

In this section we prove an interpolation result which is crucial in the proof of Theorem 1.1.

Proposition 4.1. For all εą0, there is Kpεq ”Kpε, α, dq ą0 such that

(9) }∇_v_f_}2 L2 vpµ´α1q ď Kpεq ´ S_v_p_{f, f}_{q ` }}_Π_f_}2 L2 vpµ´α1q ¯ ` ε CF Svp∇vf,∇vfq

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Proof. One can use the chain rule and an interpolation ofH9_v1 betweenH9vα{2 and H9v1`α{2 (easily

shown in Fourier variables) to get

}p∇_v_f_q_µ´_α1{2_}2 L2 v ď 2} ∇_v_p_{f µ}´_α1{2_q}2 L2 v`2}fp ∇_v_µ´_α1{2_q}2 L2 v ď Kpεq }f µ´α1{2}2₉ Hvα{2 `ε}∇_v_pf µ´α1{2q}2₉ Hvα{2 `2}fp∇_vµα´1{2q}2_L2 v ď Kpεq }f µ´α1{2}2 Hvα{2 `ε}p∇_vfqµ´α1{2}2 Hvα{2 `C}f µ´α1{2}2_L2 v ď Kpεq }f µ´α1{2}2 Hvα{2 ` ε}p∇_vfqµ´α1{2}2 Hvα{2

up to changing the value of Kpεq and where we used the fact that |p∇_vµαqµ´α1| P L8pRdq to

bound the third term. Now observe that

}f µ´_α1{2}2 Hvα{2 ď 2 ˆ }pf ´Πfqµ´_α1{2}2 Hvα{2 ` }pΠ fqµ´_α1{2}2 Hvα{2 ˙ , and that }pΠfqµ´α1{2}_Hα{2 v “ }Πf}L2vpµ´α1q}µ 1{2 α }_Hα{2 v with }µ 1{2 α }_Hα{2 v ď C since µ 1{2 α P Hv1

from (3) and (4). Moreover, one has ∇_vf “ ∇_vf ´Π∇_vf. One can conclude by using (8)

twice.

5. Proof of Theorem 1.1 Up to changing fin by fin´@finDµα, we assume that

ť

Td_ˆ_Rdfpt, x, vqdvdx “ 0 at initial

time t“0, so that by conservation it also holds for all time tą0. We introduce a new norm on the weighted Sobolev space H_x,v1 pµ´_α1q. It is defined by

(10) ~f~2 “ }f}2_L2_x,vp_µ´1 α q ` a} ∇_xf}2_L2 x,vpµ´α1q ` b} ∇_vf}2_L2_x,vp_µ´1 α q`2cx ∇_xf,∇_vfy_L2 x,vpµ´α1q,

wherea,b andcare positive constants to be determined later on. Observe that as soon asc2 _ă ab, one has that~ ¨ ~is equivalent to} ¨ }_H1

x,vpµ´α1q. Let us note that the commutatorsr∇x, v¨∇xs

and r∇_x, Lαsvanish whiler∇v, v¨∇xs “ ∇x and r∇v, Lαs “ ∇v. Also observe that v¨∇x is

skew-symmetric in L2_x,vpµ´_α1q.

Let us estimate the evolution of each term appearing in the new norm defined in (10) for f a solution of (1) with initial data fin satisfying @finD= 0. In the following the notation S_x,v

denotes the integral of S_v _{in the}x variable. One has 1 2 d dt}f} 2 L2 x,vpµ´α1q“ ´ S_x,v_pf, fq, 1 2 d dt}∇xf} 2 L2 x,vpµ´α1q“ ´ S_x,v_p∇_xf,∇_xfq, 1 2 d dt}∇vf} 2 L2 x,vpµ´α1q“ ´ S_x,v_p∇_vf,∇_vfq ` }∇_vf}2_L2 x,vpµ´α1q ´ x ∇_xf,∇_vfy_L2 x,vpµ´α1q , d dtx∇xf,∇vfyLx,v2 pµ´α1q“ ´}∇xf} 2 L2 x,vpµ´α1q´2 S_x,v_p∇_xf,∇_vfq ` x∇_xf,∇_vfy_L2 x,vpµ´α1q .

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HYPOCOERCIVITY WITH HEAVY-TAILED EQUILIBRIUM 7 Notice here that the keystone of the proof of the last equality is the splitting obtained in Proposition 2.1. By gathering all the previous estimates one gets

1 2 d dt~f~ 2 _{“ ´}_S x,vpf, fq ´aSx,vp∇xf,∇xfq ´bSx,vp∇vf,∇vfq ´c}∇xf}2_L2 x,vpµ´α1q ` b}∇_vf}2_L2 x,vpµ´α1q ´ bx ∇_xf,∇_vfy_L2 x,vpµ´α1q ´2cS_x,v_p∇_xf,∇_vfq `cx∇_xf,∇_vfy_L2 x,vpµ´α1q .

The first four terms are dissipation terms and the last four terms are remainder terms. Let us control the latter by the former ones. By integrating (5) inx and using Young’s inequality one gets |2cS_x,v_p∇_x_f,∇_v_f_{q| ď} 2c 2 b Sx,vp∇xf,∇xfq ` b 2Sx,vp∇vf,∇vfq. Then since ş∇_vfdv“0, one has

b ˇ ˇ ˇx∇_xf,∇_vfy_L2 x,vpµ´α1q ˇ ˇ ˇ “ b ˇ ˇ ˇx∇_xf´Πp∇_xfq,∇_vfy_L2 x,vpµ´α1q ˇ ˇ ˇ ď b C_P1{2S_x,v_p∇_xf,∇_xfq1{2_}_∇ vf}_L2 x,vpµ´α1q ď b CP 2 Sx,vp∇xf,∇xfq ` b 2}∇vf} 2 L2 x,vpµ´α1q,

where we used (6). Similarly c ˇ ˇ ˇx∇_xf,∇_vfy_L2_x,vp_µ´1 α q ˇ ˇ ˇ ď c 2_C P 2b Sx,vp∇xf,∇xfq ` b 2}∇vf} 2 L2 x,vpµ´α1q.

For the last remainder term we use (9) integrated in x, namely

}∇_vf}2 L2 x,vpµ´α1q ď Kpεq ´ S_x,v_pf, fq ` }Πf}2 L2 x,vpµ´α1q ¯ ` ε CFSx,vp∇vf,∇vfq.

We can use the Poincar´e inequality on the torus (sincef is mean-free) and the Jensen inequality to get }Πf}2 L2 x,vpµ´α1q ď r CP}∇xf}2_L2 x,vpµ´α1q where r

CP ” CrPpdq is the Poincar´e constant of the

d-dimensional torus. Thus eventually, one has

(11) 1

2 d dt~f~

2_`_D_p_{f, f}_{q ď}₀_, where the dissipation is given by

Dpf, fq “ p1´2b KpεqqS_x,v_pf, fq ` ˆ a´c 2 b ˆ 2`CP 2 ˙ ´bCP 2 ˙ S_x,v_p∇_xf,∇_xfq ` ˆ b 2´2bεCF ˙ S_x,v_p∇_vf,∇_vfq `´c´2bCrPKpεq ¯ }∇_xf}2_L2 x,vpµ´α1q.

Now choose consecutively ε, b, c and a such that 0 ă ε ă 1{p4CFq, 0 ă b ă 1{p2Kpεqq,

cą2bCrPKpεqand finallyalarge enough so thataąc2p2`CP{2q {b`bCP{2. It yields that the

dissipation is non-negative and even that there is a constantλą0 (depending ona, b, c, ε) such thatDpf, fq ěλ~f~2_{. By a Gronwall type argument we have that}_~_f_p_t_q~_{decays exponentially}

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6. The case of the heavy-tailed BGK equation In this last section we consider another simple kinetic model

(12) Btf`v¨∇xf “ ΠMf ´f , with pΠMfqpt, x, vq “ Mpvq ż

Rd

fpt, x, wqdw , for which the local equilibrium satisfies the following assumptions

(13) Mpvq ą0,

ż

Rd

M “1, and ∇_v_ln_p_M_{q P}_L8_.

This allows for heavy-tailed distributions, namely M such that Mpvq „|v|Ñ8 |v|´d´α with

αP p0,2q.

Theorem 6.1. Assume that (13) holds and let f solve the BGK equation (12) starting from the initial data fin

PH_x,v1 pM´1q. Then, for all tě0 one has

}fptq ´@finDM}H_x,vp1 _M´1_q ď C}f

in

´@finDM}H_x,vp1 _M´1_qe´λt

for some constant C ě1 and λą0 depending only on d and }∇_vlnpMq}L8.

The proof is similar and simpler than that of Theorem 1.1. We skip many details as the reader may go back to the proof of Theorem 1.1 in order to recover them.

Proof of Theorem 6.1. Considerf a solution to (12) with initial datafin satisfying @finD“0. Let us observe that the commutators r∇_x_{, v}_¨∇_x_s_and _r∇_x_,_Π_M_s_{vanish while}_r∇_v_{, v}_¨∇_x_{s “} ∇_x

and also r∇_v,ΠMs “ ∇vlnpMqΠM. Now with this in mind, and defining the triple norm of f

as in (10) with µα replaced by M, one gets

1 2 d dt~f~ 2 _{“ ´}}_f_´_Π Mf}2L2_x,vp_M´1_q ´a}∇_xf´ΠM∇xf}2L2 x,vpM´1q´b} ∇_vf}2 L2 x,vpM´1q´c} ∇_xf}L2 x,vpM´1q ` bx∇_vlnpMqΠMf,∇vfyL2 x,vpM´1q ´ bx∇xf,∇vfyL2x,vpM´1q ´2cx∇_xf,∇_vfyL2 x,vpM´1q`cx ∇_v_ln_pMqΠMf,∇xfyL2 x,vpM´1q .

First, we notice that x∇_xf,∇_vfy_L2

x,vpM´1q “ x

∇_xf´ΠM∇xf,∇vfyL2

x,vpM´1q to deal with the

third and fourth remainder terms with Cauchy-Schwarz inequality. The last remainder term requires some special care. Indeed, observe that sincex∇_v_ln_pMqΠMf,ΠMgyvanishes for any g,

one thus has

x∇_v_ln_p_M_q_Π_M_f,∇_x_f_y L2

x,vpM´1q ď }∇_v_ln_p_M_q}_L8}Π_Mf}_L2

x,vpM´1q}∇xf ´ΠM∇xf}L2x,vpM´1q.

We also have that

x∇_v_ln_pMqΠMf,∇vfy_L2

x,vpM´1q ď }

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HYPOCOERCIVITY WITH HEAVY-TAILED EQUILIBRIUM 9 Finally, we recall that}ΠMf}L2

x,vpM´1qďCrP}∇xf}L2x,vpM´1q withCrP the Poincar´e constant of

the d-dimensional torus. Then using four times Young’s inequality with well chosen weights, one obtains (11) with the dissipation

Dpf, fq “ }f´ΠMf}2_L2 x,vpM´1q` ` a´b´4c2{b´CMc{2 ˘ }∇_x_f _´_Π_M∇_x_f_}2 L2 x,vpM´1q ` pb´b{4´b{4´b{4q }∇_vf}2_L2 x,vpM´1q` pc´bCM ´c{2q } ∇_xf}L2 x,vpM´1q

with CM “ }∇vlnpMq}2L8Cr_P2. One concludes as in Theorem 1.1 after choosing any b ą 0,

cą2b CM and finally aą4c2{b`b`Cd,Mc{2.

Acknowledgements

Maxime Herda thanks the LabEx CEMPI (ANR-11-LABX-0007-01). H´el`ene Hivert thanks the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC starting grant MESOPROBIO no_{639638). Isabelle Tristani thanks} the ANR EFI: ANR-17-CE40-0030 and the ANR SALVE: ANR-19-CE40-0004 for their support. This work is part of a collaborative research project that was initiated for the Junior Trimester Program in Kinetic Theory at the Hausdorff Research Institute for Mathematics in Bonn. Part of the work was carried out during the time spent at the institute. The authors are grateful for this opportunity and warmly acknowledge the HIM for the financial support and the hospitality they benefited during their stay.

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(N. Ayi)Sorbonne Universit´e, Universit´e de Paris, CNRS, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France

E-mail address: [email protected]

(M. Herda) Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlev´e, F-59000 Lille, France

(H. Hivert) Univ. Lyon, ´Ecole centrale de Lyon, CNRS UMR 5208, Institut Camille Jordan, F-69134 ´Ecully, France

(I. Tristani) Département de mathématiques et applications, École normale supérieure, CNRS, PSL Research University, 45 rue d’Ulm, 75005 Paris, France