The transport operatorT on the phase space (with positionxand velocityv) can be rewritten asT=i v·ξfˆin the Fourier variable ξassociated to x

(1)

L² hypocoercivity, inequalities and applications Jean Dolbeault

Let us consider the kinetic equation

(1) ∂tf +Tf =Lf

where Lf is either the Fokker-Planck operator L₁f =∇_v· M ∇_v(M⁻¹f) or a scattering collision operatorLf =R

R^dσ(·, v⁰) f(v⁰)M(·)−f(·)M(v⁰)

dv⁰, for in- stance the simplest possible one, the linear BGK operatorL2f =ρM(v)−f where ρ(t, x) =R

R^df(t, x, v)dvis the spatial density andM(v) = (2π)^−d/2e^−|v|²^/2. The transport operatorT on the phase space (with positionxand velocityv) can be rewritten asT=i v·ξfˆin the Fourier variable ξassociated to x. With the op- erators Π andA defined respectively by Π ˆf :=MR

R^d

fˆ(ξ, w)dw andAfˆ(ξ, v) :=

−i ξ(1+|ξ|²)⁻¹·R

R^dwfˆ(ξ, w)dwM(v), the L²entropy, or L²Lyapunov functional H[ ˆf] := ¹₂kfˆk²+δRehAf ,ˆfˆiwherekgk²:=RR

R^d|g(w)|²dγ,dγ(w) :=M(w)⁻¹dw, is such that, iff solves (1), then theentropy – entropy production inequality

d

dtH[ ˆf(t, ξ,·)]≤ −λH[ ˆf(t, ξ,·)]

holds if Q(X, Y) :=

1−_1+|ξ|^δ^|ξ|²₂ −^λ₂

X²−_1+|ξ|^δ^|ξ|₂ 1 +√

3|ξ|+λ

X Y + _δ

|ξ|² 1+|ξ|² −^λ₂

Y² is a nonnegative quadratic form of X and Y, where X :=k(1−Π) ˆfk and Y :=

kΠ ˆfk. Here it is clear thatξ∈R^d can be considered as a parameter, that is, we can perform amode-by-mode analysis. Proving the exponential decay of H[ ˆf] for some δ = δ(|ξ|) with a rate λ =λ(|ξ|) is reduced to the discriminant condition which guarantees thatQ≥0. It turns out thatH[ ˆf] is equivalent tokfˆk²ifδ <2 and one can prove by the method of [9, 5] that

kf(t, ξ,ˆ ·)k²≤C(|ξ|)kfˆ0(ξ,·)k²e^−λ(|ξ|)^t ∀t≥0, ξ∈R^d,

where C(|ξ|) = (2 +δ(|ξ|))/(2−δ(|ξ|)). This has been analysed in [5] in terms of asymptotic decay rates of kf(t,·,·)k²_L₂₍

R^d×R^d,dx dγ) for a choice of δ which is independent ofξ but can be refined by taking aξ-dependent value ofδ.

Theorem 1. [2]Iff solves (1)withL=L1orL=L2for some nonnegative initial datum f0∈L²(R^d×R^d, dx dγ)∩L² R^d, dγ; L¹(R^d, dx)

, then

kf(t,·,·)k²_L2(R^d×R^d,dx dγ)≤(2π)^−dΨM,Q(t) ∀t≥0 withM =kf₀k_L2(R^d,dγ;L¹(R^d,dx)),Q=kf₀k_L2(R^d×R^d,dx dγ), and Ψ_M,Q(t) := inf_R>0

RR

0 C(s)e⁻^λ(s)^ts^d−1ds ω_dd M²+ sup_s≥RC(s)e⁻^λ(R)tQ² .

The proof of this result is reminiscent of the proof in [11] ofNash’s inequality (2) kuk²⁺_L₂₍⁴^d

R^d)≤ C_Nashkuk_L⁴^d₁₍

R^d)k∇uk²_L2(R^d)

(2)

and the definition of the operatorAis inspired by the diffusion limit: see [9, 2]. It is well known that a solution to the heat equation

∂tu= ∆u ∀(t, x)∈R⁺×R^d decays according to

ku(t,·)k²_L2(R^d,dx)≤ C ku₀k²_L2(R^d,dx)(1 +t)⁻^d²

after computing _dt^dku(t,·)k_L2(R^d,dx) = −2k∇u(t,·)k_L2(R^d,dx) and taking (2) into account. See [7] for a discussion of the optimality of such an estimate. This decay rate can be recovered also at the kinetic level for the solution of (1): see [5].

The next question is of course to understand what happens in presence of an external potential. Let us start at diffusive level with the Fokker-Planck equation (3) ∂tu= ∆u+∇ ·(u∇V) ∀(t, x)∈R⁺×R^d

whereV is a given external potential. To fix ideas, we shall assume that V(x) =

|x|^αfor someα >0 and discuss the cases depending on the value ofα.

BForα≥1, we have thePoincar´e inequality Z

R^d

|v−v|¯²dµα≤ C Z

R^d

|∇v|²dµα

with ¯v :=R

R^dv dµα and dµα :=Uα(x)dx,Uα(x) := Z_α⁻¹e^−|x|^α,Zα:=R

R^dUαdx.

It is then standard to prove that a solutionu(t,·) of (3) is such thatv :=u/Uα

satisfies Z

R^d

|v(t,·)−v|¯²dµα≤ Z

R^d

|v(0,·)−v|¯²dµαe⁻²^C^t ∀t≥0.

BThe caseα∈(0,1) has been studied in [10] using the weak Poincar´e inequality.

This approach requires the existence of a uniform bound. Alternatively, we can consider theweighted Poincar´e inequality

(4)

Z

R^d

|∇v|²dµ_α≥ C Z

R^d

|v−v|¯²hxi^−βdµ_α

withβ = 2 (1−α) and the same notations as above for ¯v and dµα. Here we use the notationhxi=p

1 +|x|² and notice thatβ vanishes as α→1₋. In order to compensate for the additional weight in the right hand side in (4), it is convenient to introduce a weighted L² norm with a weighthxi^k.

Theorem 2. [4] Assume that α∈(0,1). If usolves (3) with initial datumu₀∈ L¹₊(R^d, dµ_α)∩L²(R^d,hxi^kdµ_α) for somek >0,v=u/U_α andv₀=u₀/U_α, then

Z

R^d

|v(t,·)−v|¯² dµα≤ Z

R^d

|v0−v|¯² dµα

−β/k

+Kt ^−k/β

∀t≥0,

for some constantK depending onk andu₀.

2

(3)

BIn the limit case asα→0+, it makes sense to considerV(x) =γ log|x|. In the rangeγ∈(0, d), (3) admits no stationary solution in L¹(R^d). In that case, we can again introduce weights and consider theCaffarelli-Kohn-Nirenberg inequality

R

R^d|x|^γu²dx≤ C R

R^d|x|^−γ |∇(|x|^γu)|² dxa

R

R^d|x|^k|u|dx2 (1−a)

which generalizes Nash’s inequality (2). A decay result goes as follows.

Theorem 3. [6]Let d≥1 andγ∈(0, d), k≥max{2, γ/2}. If usolves (3) with initial datum u₀∈L¹₊(R^d,hxi^kdµ_α)∩L²(R^d, dµ_α), then there is a constantc >0 depending onu₀ such that

ku(t,·)k²_L2(R^d,|x|^γdx)≤ ku₀k²_L2(R^d,|x|^γdx)(1 +c t)⁻^d−γ² ∀t≥0.

Similar results can be obtained at kinetic level when the transport operator is defined byTf =v· ∇xf− ∇xV · ∇vf. WithL=L1 or L=L2, and appropriate estimates involvinghxi^k, results are obtained which are all consistent with a diffusion limit given by (3) and rely on the same functional inequalities. So far we have considered only Maxwellian local equilibria, but a similar discussion can be done whenM(v) =Z_β⁻¹ exp(−|v|^β) for someβ >0, depending whetherβ≥1 or not, and the caseF(v) = hvi^−γ has also been studied. Notice that a fractional diffusion limit has to be considered whenR

R^d|v|²F(v)dvis infinite. The method also adapts to equations with a Poisson coupling. See [6, 4, 8, 3, 1] for detailed statements.

References

[1] L. Addala, J. Dolbeault, X. Li, and M. L. Tayeb, L²-hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system. Preprint hal-02299535

& arXiv: 1909.12762, Sep. 2019.

[2] A. Arnold, J. Dolbeault, C. Schmeiser, and T. W¨ohrer,Sharpening of decay rates in Fourier based hypocoercivity methods. Preprint hal-03078698 & arXiv: 2012.09103, Dec.

2020.

[3] E. Bouin, J. Dolbeault, L. Lafleche, and C. Schmeiser, Fractional hypocoercivity.

Preprint hal-02377205 & arXiv: 1911.11020, Nov. 2019.

[4] E. Bouin, J. Dolbeault, L. Lafleche, and C. Schmeiser, Hypocoercivity and sub- exponential local equilibria, Monatshefte f¨ur Mathematik, 194 (2020), pp. 41–65.

[5] E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot, and C. Schmeiser,Hypocoercivity without confinement, Pure and Applied Analysis, 2 (2020), pp. 203–232.

[6] E. Bouin, J. Dolbeault, and C. Schmeiser,Diffusion and kinetic transport with very weak confinement, Kinetic & Related Models, 13 (2020), pp. 345–371.

[7] E. Bouin, J. Dolbeault, and C. Schmeiser, A variational proof of Nash’s inequality, Rendiconti Lincei – Matematica e Applicazioni, 31 (2020), pp. 211–223.

[8] C. Cao, the kinetic Fokker-Planck equation with weak confinement force. Preprint hal- 01697058 & arXiv: 1801.10354, June 2018.

[9] J. Dolbeault, C. Mouhot, and C. Schmeiser,Hypocoercivity for linear kinetic equations conserving mass, Transactions of the American Mathematical Society, 367 (2015), pp. 3807–

3828.

[10] O. Kavian, S. Mischler, and M. Ndao,The Fokker-Planck equation with subcritical confinement force. Preprint hal-01241680 & arXiv: 1512.07005, Dec. 2016.

[11] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), pp. 931–954.

3