CONFINEMENT
Emeric Bouin∗
CEREMADE (CNRS UMR n◦7534), PSL university
Universit´e Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
Jean Dolbeault
CEREMADE (CNRS UMR n◦7534), PSL university
Universit´e Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
Christian Schmeiser
Fakult¨at f¨ur Mathematik, Universit¨at Wien Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
Abstract. This paper is devoted to Fokker-Planck and linear kinetic equa- tions with very weak confinement corresponding to a potential with an at most logarithmic growth and no integrable stationary state. Our goal is to understand how to measure the decay rates when the diffusion wins over the confinement although the potential diverges at infinity. When there is no con- finement potential, it is possible to rely on Fourier analysis and mode-by-mode estimates for the kinetic equations. Here we develop an alternative approach based on moment estimates and Caffarelli-Kohn-Nirenberg inequalities of Nash type for diffusion and kinetic equations.
1. Introduction. This paper addresses the large time behavior of the solutions to the macroscopic Fokker-Planck equation and to kinetic equations with Fokker- Planck or scattering collision operators.
The first part of this paper deals with themacroscopic Fokker-Planck equation
∂u
∂t = ∆xu+∇x·(∇xV u) =∇x e−V ∇x eV u
(1) where x ∈ Rd, d ≥ 3, and V is a potential such that e−V 6∈ L1(Rd), that is, e−V dx is an unbounded invariant measure. There are various reasons to consider only dimensions larger than 3, among which the use of the Hardy inequality. In some cases, the dimensiond= 2 is also covered as a limit case, while estimates in dimensiond= 1 are of different nature and will not be considered in this paper for sake of simplicity. We shall investigate the two following examples
V1(x) =γ log|x| and V2(x) =γ loghxi
Date: September 8, 2019.
2010Mathematics Subject Classification. Primary: 35B40, 35Q84; Secondary: 82C40, 76P05, 26D10.
Key words and phrases. Nash’s inequality; Caffarelli-Kohn-Nirenberg inequalities; decay rates;
semigroup; weak Poincar inequality; unbounded invariant measure; rate of convergence; Fokker- Planck operator; kinetic equations; scattering operator; transport operator; hypocoercivity.
∗Corresponding author: Emeric Bouin.
with γ < d and hxi:=p
1 +|x|2 for anyx∈Rd. These two potentials share the same asymptotic behavior as|x| → ∞. The potentialV1is invariant under scalings, whereasV2 is smooth at the origin. In both cases, the only integrable equilibrium state is 0. Thus, if the initial datum u0 is such thatu0 ∈L1(Rd), we expect that the solution to (1) converges to 0 ast→+∞. Whenγ >0, the potentialV isvery weakly confiningin the sense that, even if it eventually slows down the decay rate, it is not strong enough to produce a stationary state of finite mass: the diffusion wins over the drift. Our goal to establish the rate of convergence in suitable norms. We shall use the notation k·kp := k·kLp(dx) in case of Lebesgue’s measure and specify the measure otherwise.
Theorem 1.1. Assume that either d≥3, γ <(d−2)/2 and V =V1 orV =V2, or d = 2, γ ≤ 0 and V = V2. Then any solution u of (1) with initial datum u0∈L1+∩L2(Rd)satisfies, for allt≥0,
ku(t,·)k22≤ ku0k22
(1 +c t)d2 with c:= 4 d minn
1,1−d−22γ o
CNash−1 ku0k4/d2
ku0k4/d1 . (2) HereCNash denotes the optimal constant in Nash’s inequality [24,11]
kuk2+2 d4 ≤ CNash kuk1d4k∇uk22 ∀u∈L1∩ H1(Rd). (3) Note that the rate of decay is independent ofγand we recover the classical estimate due to J. Nash when V = 0 (here γ = 0). The proof of Theorem 1.1 and further considerations on optimality are collected in Section2.1. Our method involves the computation of ∆V. In dimension d = 2, V = V1 would produce a singularity (which could be handled by an appropriate regularization procedure). In dimension d = 1, V200(x) = (1−x2)/(1 +x2)2 has no definite sign and would require new estimates, which are not covered by our result.
Theorem1.1does not cover the interval (d−2)/2< γ < d. This range is covered by employing the natural setting of L2 eV
and by requiring additional moment bounds.
Theorem 1.2. Let d≥1, γ < d, V =V1 or V =V2, and u0∈ L1+∩L2 eV . If γ > 0, let us assume that
|x|ku0
1 < ∞ for some k ≥ max{2, γ/2}. Then any solution of (1)with initial datumu0 satisfies
∀t≥0, ku(t,·)k2L2(eVdx)≤ ku0k2L2(eVdx)(1 +c t)−d−γ2 . The constantc depends ond,γ,k,ku0kL2(eVdx),ku0k1, and
|x|ku0
1.
The proof of Theorem1.2is done in Section 2.2. Although this is a side result, let us notice that the case in which the potential contributes to the decay,i.e., when γ < 0, is also covered in Theorem 1.2. The scale invariance of (1) with V = V1
can be exploited to obtain intermediate asymptotics in self-similar variables. Let us define
u?(t, x) = c?
(1 + 2t)d−γ2 |x|−γexp
− |x|2 2 (1 + 2t)
, (4)
The following result on intermediate asymptotics allows us to identify the leading order term of the solution of (1) ast→+∞. It is the strongest of our results on (1) but initial data need to have a sufficient decay as|x| → ∞.
Theorem 1.3. Let d≥1,γ∈(0, d)andV =V1. If for some constantK >1, the function u0 is such that
∀x∈Rd, 0≤u0(x)≤K u?(0, x)
where c? is chosen such thatku?k1=ku0k1 then the solution uof (1) with initial datum u0 satisfies
∀t≥0, ku(t,·)−u?(t,·)kp≤K c1−
1
? pku0k
1 p
1
e 2|γ|
γ2 1−1 p
(1 + 2t)−ζp for any p∈[1,+∞), whereζp:=d2 1−1p
+21p min
2,d−γd .
More detailed results will be stated in Section2.3. Let us quote some relevant papers for (1). In the case without potential, the decay rates of the heat equation is known for more than a century and goes back to [16]. Standard techniques use the Fourier transform, Green kernel estimates and integral representations: see for instance [15]. There are many other parabolic methods which provide decay rates and will not be reviewed here like, for instance, the Maximum Principle, Harnack inequalities and the parabolic regularity theory: see for instance [28].
In his celebrated paper [24], J. Nash was able to reduce the question of the decay rates for the heat equation to (3): see [8] for detailed comments on the optimality of such a method. Entropy methodshave raised a considerable interest in the recent years, but the most classical approach based on the so-calledcarr´e du champ method applies to (1) only for potentialsV with some convexity properties and a sufficient growth at infinity: typically, if V(x) = |x|α, then α≥1 is required for obtaining a Poincar´e inequality and the rate of convergence to a unique stationary solution is then exponential, when measured in the appropriate norms; see [4] for a general overview. An interesting family ofweakly confiningpotentials is made of functionsV with an intermediate growth, such thate−V is integrable but lim|x|→∞V(x)/|x|= 0:
all solutions of (1) are attracted by a unique stationary solution, but the rate is expected to be algebraic rather than exponential. A typical example isV(x) =|x|α withα∈(0,1). The underlying functional inequality is aweak Poincar´e inequality:
see [26, 21], and [3] for related Lyapunov type methods `a la Meyn and Tweedie or [6] for recent spectral considerations. We refer to [2] and [30,31, 32] for further considerations on, respectively, weighted Nash inequalities and spectral properties of the diffusion operator. This problem has also attracted attention in the physics literature (see [1] and the references therein for a list of interesting examples).
The second part of this paper is devoted tokinetic equations involving a degen- erate diffusion operator acting only on the velocity variable or scattering operators, for very weak potentials like V1 or V2. Various hypocoercivity methods have been developed over the years in, e.g., [17, 18, 23, 29, 13], in order to prove exponen- tial rates in appropriate norms, in presence of a strongly confining potential. In that case, the growth of the potential at infinity has to be fast enough not only to guarantee the existence of a stationary solution but also to provide macroscopic coercivity properties which typically amount to a Poincar´e inequality. A popular simplification is to assume that the position variable is limited to a compact set, for example a torus. Such results are the counterpart in kinetic theory of diffusions covered by thecarr´e du champ method, as emphasized in [5].
Recently, hypocoercivity methods have been extended in [7] to the case without any external potential by replacing the Poincar´e inequality by Nash type estimates.
Thesub-exponential regime or the regime with weak confinement, i.e., of a poten- tialV such that a weak Poincar´e inequality holds, has also been studied in [10,19].
What we will study next is the range ofvery weak potentialsV, which have a growth at infinity which is below the range of weak Poincar´e inequalities, but are still such that lim|x|→∞V(x) = +∞. This regime is the counterpart at kinetic level of the results of Theorems1.1,1.2and 1.3. As in the case of (1) whenγ≥0, the drift is opposed to the diffusion, but it is not strong enough to prevent that the solution locally vanishes.
Let us consider the kinetic equation
∂tf+v· ∇xf− ∇xV · ∇vf =Lf (5) whereLf is one of the two following collision operators:
(a) a Fokker-Planck operator Lf =∇v·
M∇v M−1f , (b) a scattering collision operator
Lf = Z
Rd
σ(·, v0) f(v0)M(·)−f(·)M(v0) dv0. We consider the case of a global equilibrium of the form
∀(x, v)∈Rd×Rd, M(x, v) =M(v)e−V(x) where M(v) = (2π)−d2e−12|v|2. We shall say that the gaussian function M(v) is the local equilibrium and assume that thescattering rate σ(v, v0) satisfies
(H1) 1≤σ(v, v0)≤σ , ∀v , v0 ∈Rd, for some σ≥1, (H2)
Z
Rd
σ(v, v0)−σ(v0, v)
M(v0)dv0= 0 ∀v∈Rd.
Notice thatM 6∈L1(Rd×Rd) ifV =V1orV =V2, so that the space L2 M−1dx dv is defined with respect to an unbounded measure. As in the case of (1), the only integrable equilibrium state is 0. Thus, if the initial datum f0 is such that f0∈L1(dx dv), we expect that the solution to (5) converges to 0 locally ast→+∞
and look for the rate of convergence in suitable norms.
WhenV = 0, the optimal rate of convergence of a solution f of (5) with initial datum f0 is known. In [7], it has been proved that there exists a constant C >0 such that
Z Z
Rd×Rd
|f(t,·,·)|2 dµ≤C(1 +t)−d2 Z Z
Rd×Rd
|f0|2 dµ ∀t≥0,
wheredµ=M−1dx dv and by factorization, the result is extended with same rate for an arbitrary ` > d to the measure hvi`dx dv if f0 ∈L2 Rd×Rd,hvi`dx dv
∩ L2+ Rd,hvi`dv; L1 Rd, dx
. Our main result on (5) is a decay rate in the presence of a very weak potential. It is an extension of the results of Theorem 1.2 to the framework of kinetic equations.
Theorem 1.4. Let d ≥ 1, V = V2 with γ ∈ [0, d) and k > max{2, γ/2}. We assume that (H1)–(H2) hold and consider a solution f of (5) with initial datum f0∈L2(M−1dx dv) such that RR
Rd×Rdhxikf0dx dv+RR
Rd×Rd|v|kf0dx dv < +∞.
Then there existsC >0 such that
∀t≥0, kf(t,·,·)k2L2(M−1dx dv)≤C(1 + t)−d−γ2 .
Standard methods of kinetic theory can be used to establish the existence of solutions of (5) whenV =V2. We will not give details here. At formal level, similar results can be expected whenV =V1 but the singularity atx= 0 raises difficulties which are definitely out of the scope of this paper.
The expression of the constant C is explicit. However, due to the method, we cannot claim optimality in the estimate of Theorem1.4, but at least the asymptotic rate is expected to be optimal by consistency with the diffusion limit, as it is the case when V = 0, studied in [7]. The strategy of the proof and further relevant references will be detailed in Section3.
2. Decay estimates for the macroscopic Fokker-Planck equation. In this section, we establish decay rates for (1) and discuss the optimal range of the pa- rameters.
2.1. Decay inL2(Rd). We prove Theorem1.1. By testing (1) withu, we obtain d
dt Z
Rd
u2dx=−2 Z
Rd
|∇u|2dx+ Z
Rd
∆V|u|2dx , with eitherV =V1 orV =V2and
∆V1(x) =γd−2
|x|2 and ∆V2(x) =γ d−2
1 +|x|2 + 2γ (1 +|x|2)2. Forγ≤0, with the restriction that V =V2 ifd= 2, we deduce
d
dtkuk22≤ −2k∇uk22≤ − 2 CNash
ku0k−4/d1 kuk2+4/d2 ,
from Nash’s inequality (3). Integration completes the proof of (2). For the case 0< γ <(d−2)/2 we use the following Hardy-Nash inequalities.
Lemma 2.1. Let d≥3andδ <(d−2)2/4. Then kuk2+2 4d ≤ Cδ
k∇uk22−δ Z
Rd
u2
|x|2dx
kuk14d ∀u∈L1∩H1(Rd), (6) with
Cδ=CNash
1−(d−2)4δ 2
−1
.
Let additionally η <(d2−4)/4. Then, for any u∈L1∩ H1(Rd), kuk2+2 4d ≤ Cδ,η
k∇uk22−δ Z
Rd
u2
hxi2dx−η Z
Rd
u2 hxi4dx
kuk14d (7) with
Cδ,η=CNash
minn
1−(d−2)4δ 2,1−d24−4η
o−1 .
The proof of Lemma 2.1 is given in Appendix C. We use Lemma 2.1 with δ= γ(d−2)/2 and withη=γ (forV =V2), and proceed as forγ≤0 to complete the proof of Theorem1.1.
Remark 2.2. The conditionδ <(d−2)2/4 in Lemma2.1is optimal for (6) and (7).
Ifd≥3, the restriction onγin Theorem1.1is also optimal. Letd≥3,γ >(d−2)/2 andV =V1orV =V2. Then there existsu∈L1∩H1(Rd) such that kuk2= 1 and
−2 Z
Rd
|∇u|2dx+ Z
Rd
∆V|u|2dx >0.
In the case V =V1, it is indeed enough to observe that (d−2)2/4 is the optimal constant in Hardy’s inequality (see AppendixC). The caseV =V2follows from the caseV =V1 by an appropriate scaling.
2.2. Decay inL2(eV dx). By testing (1) withu eV, we obtain 1
2 d dt
Z
Rd
u2eV dx=− Z
Rd
e−V
∇ u eV
2dx . (8)
In the caseV =V1, we haveeV =|x|γ and (8) takes the form 1
2 d dt
Z
Rd
|x|γu2dx=− Z
Rd
|x|−γ |∇(|x|γu)|2dx .
We first prove Theorem1.2forγ≤0. Withγ≤0 anda= d+2−γd−γ , the inequality Z
Rd
|x|γu2dx≤ C Z
Rd
|x|−γ |∇(|x|γu)|2dx aZ
Rd
|u|dx 2(1−a)
(9) follows from the Caffarelli-Kohn-Nirenberg inequalities (see AppendixA, Ineq. (26) applied withk= 0 tov=|x|γu). The conservation of the L1 norm ofugives
d dt
Z
Rd
|u|2|x|γdx≤ −2C−(1+d−γ2 )ku0k−
4 d−γ
1
Z
Rd
|u|2|x|γdx
1+ 2 d−γ
. The conclusion of Theorem 1.2 follows by integration. An analogous argument based on the inhomogeneous Caffarelli-Kohn-Nirenberg inequality
Z
Rd
|u|2hxiγdx≤ K Z
Rd
hxi−γ |∇(hxiγu)|2dx aZ
Rd
|u|dx 2(1−a)
with a= d−γ d+ 2−γ applies to the caseγ≤0,V =V2 (see Appendix B, Ineq. (31) applied withk= 0 and v = hxiγu) if γ ≤ 2 (d−2). A minor modification (based on Appendix B, Ineq. (30)) allows us to deal with the remaining cases.
Without additional assumptions, it is not possible to expect a similar result for γ >0. Let us explain why. In the caseV =V1 and withv=|x|γu, let us consider the quotient
Q[v] :=
R
Rd|x|−γ|∇v|2dxa R
Rd|x|−γ|v|dx2(1−a)
R
Rd|x|−γv2dx
As a consequence of (9),Q[v] is bounded from below by a positive constant ifγ≤0 anda= (d−γ)/(d−γ+ 2). Let us consider the caseγ >0.
Lemma 2.3. Letd≥1,γ∈(0, d)anda= (d−γ)/(d−γ+2). Then there exists a se- quence(vn)n∈Nof smooth, compactly supported functions such that lim
n→∞Q[vn] = 0.
Proof. Let us take a smooth functionv and considervn(x) =v(x+ne) for some e∈Sd−1. ThenQ[vn] =O n−(1−a)γ
. Withγ >0, we know thatais in the range 0< a <1 if and only ifγ∈(0, d).
For the proof of Theorem 1.2 in the case 0 < γ < d, V = V1, we start by estimating the growth of the moment
Mk(t) :=
Z
Rd
|x|ku dx ,
which evolves according to Mk0 =k d+k−2−γ
Z
Rd
u|x|k−2dx≤k d+k−2−γ M
2 k
0 M1−
2 k
k ,
where we have used H¨older’s inequality andM0(t) =M0(0) =ku0k1. Integration gives
Mk(t)≤
Mk(0)2/k+ 2 d+k−2−γ
M02/ktk/2 .
Ifγ∈(0, d) anda= d+2k+2−γd+2k−γ , by inserting the Caffarelli-Kohn-Nirenberg inequal- ity (26) (see Appendix A) applied tov=|x|γu, that is
Z
Rd
|x|γu2dx≤ C Z
Rd
|x|−γ |∇(|x|γu)|2dx aZ
Rd
|x|k|u|dx 2(1−a)
, in (8), we observe that the functionz=R
Rdu2|x|γdxsolves dz
dt ≤ −2 C−1z1+d+2k−γ2
Mk(t)−d+2k−γ4 , and, after integration,
z(t)≤z(0) 1 +a
1 +b t1−d+2k−γ2k
−1−d+2k−γ2
with aand b depend only on the quantities entering into the constantc of Theo- rem1.2. Letθ= 2k/(d+ 2k−γ) and observe that
1 +a
1 +b t1−θ
−1
≥ 1 +c t1−θ
∀t≥0, ifc=bmin
a, a1/(1−θ) . Our estimate becomes z(t)≤z(0)
1 +a
1 +b t1−θ
−1−k/θ
≤z(0) 1 +c t−k(1−θ)/θ
=z(0) 1 +c t−d−γ2
. In the caseV =V2we can adopt the same strategy, based on a moment now defined as
Mk(t) :=
Z
Rd
hxiku dx ,
and on the inhomogeneous Caffarelli-Kohn-Nirenberg inequality Z
Rd
hxiγu2dx≤ K Z
Rd
hxi−γ|∇(hxiγu)|2dx a
Mk2(1−a)
with a= d+ 2k−γ d+ 2 + 2k−γ (see AppendixB, Ineq. (31) applied tov =hxiγu) ifγ≤2 (d−2). Again a minor modification (based on AppendixB, Ineq. (30)) allows us to deal with the remaining cases. This completes the proof of Theorem1.2.
2.3. Decay in self-similar variables and intermediate asymptotics.
We prove Theorem1.3. With the parabolic change of variables u(t, x) = (1 + 2t)−d/2v(τ, ξ), τ =12 log(1 + 2t), ξ= x
√1 + 2t, (10) which preserves mass and initial data, (1) is changed into
∂v
∂τ = ∆ξv+∇ξ·(v∇ξΦ), (11) where
Φ(τ, ξ) =V (eτξ) +12|ξ|2.
We investigate the long-time behavior of solutions of (1) by considering quasi- equilibria
v?(τ, ξ) :=M(τ)e−Φ(τ,ξ), (12) of (11) with an appropriately chosenM(τ).
For the scale invariant caseV =V1, the potential Φ1(τ, ξ) =γ log|ξ|+τ +12|ξ|2 in (11) can be replaced by the time independent potentialφ1(x) =γ log|ξ|+12|ξ|2. WithM(τ) =c?eγ τ, time independent equilibria
v?,1(ξ) :=c?|ξ|−γe−|ξ|2/2, (13) are available. For the second caseV =V2 with potential
Φ2(τ, ξ) := γ2 log 1 +e2τ|ξ|2
+12|ξ|2, we shall use
v?,2(τ, ξ) :=c? e−2τ+|ξ|2−γ/2
e−|ξ|2/2, (14) so thatv?,2is asymptotically equivalent tov?,1 asτ→ ∞.
If a quasi-equilibrium of the form (12) satisfies
∂v?
∂τ ≥0,
which holds for both examples (13) and (14) ifγ >0, thenv? is obviously a super- solution of (11), thus proving the following result on uniform decay estimates.
Proposition 2.4. Letγ∈(0, d)andu(t, x)be a solution of (1)with initial datum such that, for some constant c?>0,
0≤u(0, x)≤c? σ+|x|2−γ/2 exp
−|x|2 2
∀x∈Rd, withσ= 0if V =V1 andσ= 1if V =V2. Then
0≤u(t, x)≤ c?
(1 + 2t)d−γ2 σ+|x|2−γ/2
exp
− |x|2 2 (1 + 2t)
∀x∈Rd, t≥0. For 0< γ < d, we obtain a pointwise decay: the attracting potential is too weak for confinement (no stationary state can exist, at least among L1(Rd) solutions) but it slows down the decay compared to solutions of the heat equation (that is, solutions corresponding toV = 0).
The result of Proposition2.4 is also true for γ ≤0 if V =V1. In that case, a repulsive potential withγ <0 accelerates the pointwise decay, but does not change the uniform decay rate ast→+∞because
∀t >0, max
r>0 r−γ exp
−r2 4t
= e
2|γ|t γ/2
. (15)
In order to obtain an estimate in L2 eVdx
, let us state a result on a Poincar´e inequality. We introduce the notations
Φγ,σ(ξ) := 12|ξ|2+γ2 log σ+|ξ|2 ,
Zγ,σ :=
Z
Rd
e−Φγ,σ(ξ)dξ and dµγ,σ:=Zγ,σ−1e−Φγ,σdξ .
Lemma 2.5. Assume thatd≥1,γ∈(0, d)andσ∈R+. With the above notations, there is a positive constant λγ,σ such that
Z
Rd
|∇w|2dµγ,σ≥λγ,σ
Z
Rd
|w−w|2dµγ,σ
∀w∈H1(Rd, dµγ,σ)such thatw= Z
Rd
w dµγ,σ. (16) Moreover, for any γ∈(0, d),minσ∈[0,1]λγ,σ>0.
Proof. Let us consider a potential ψ on Rd. We assume that ψ is a measurable function such that
`= lim
r→+∞inff∈D(Bc r)\{0}
R
Rd |∇f|2+ψ|f|2 dξ R
Rd|f|2dξ >0, whereBcr:=
x∈Rd : |x|> r and D(Bcr) denotes the space of smooth functions onRdwith compact support inBcr. According to Persson’s result [25, Theorem 2.1], either the lower end of the continuous spectrum of the Schr¨odinger operator−∆+ψ is ` <+∞, or` = +∞ and −∆ +ψ has pure discrete spectrum and λγ,σ is the lowest positive eigenvalue.
With the change of unknown function w = f eΦγ,σ/2, the problem of the best constant in (16) is transformed into the Schr¨odinger eigenvalue problem for the potential ψ = 14|∇Φγ,σ|2−12∆Φγ,σ, whose kernel is generated bye−Φγ,σ/2, from which we deduce the existence of a constant λγ,σ > 0 because ` = +∞ in that case.
In the special caseσ= 0, it is possible to computeλγ,0 as follows.
Lemma 2.6. If d≥1andγ∈(0, d), thenλγ,0= min
2,d−γd .
Proof. Since µγ,0 is radially symmetric, we can use a decomposition in spherical harmonics in order to computeλγ,0. The equality case is achieved either by a non- constant radial function, or by a function w(x) = x1f(|x|), where w solves the eigenvalue problem
−µ−1γ,0∇ ·(µγ,0∇w) =λ w .
In the first case, the problem is solved byw(x) =|x|2−d+γ andλ= 2, while in the second case the problem is solved byf ≡1 andλ=d/(d−γ).
An interesting consequence of Lemma2.6is a result ofintermediate asymptotics, which allows us to identify the leading order term of the solution of (1) ast→+∞.
Corollary 2.7. Assume that d≥1,γ∈(0, d)andV =V1. With the above nota- tions, ifusolves (1)with an initial datumu0∈L1+(Rd)such that u?(0, x)−1
u20∈
L1+(Rd), with u? defined by (4), and if we choosec? in (4) such that ku?(0,·)k1= ku0k1, then
Z
Rd
u(t, x)−u?(t, x)2
u?(t, x) dx≤(1 + 2t)−λγ,0 Z
Rd
u(0, x)−u?(0, x)2
u?(0, x) dx . Proof. By definition ofu?, we have
Z
Rd
v?,1dξ = Z
Rd
v(0, ξ)dξ= Z
Rd
u0dx .
Then, using the Poincar´e inequality (16) and Lemma2.6, we know that d
dτ Z
Rd
(v−v1,?)2eφ1dξ=−2 Z
Rd
∇ξ eφ1(v−v1,?)
2e−φ1dξ
≤ −2λγ,0
Z
Rd
(v−v1,?)2eφ1dξ , from which we deduce that
Z
Rd
(v−v1,?)2eφ1dξ≤e−2λγ,0τ Z
Rd
(u(0, x)−v1,?)2eφ1dx . This concludes the proof using the parabolic change of variables (10).
Proof of Theorem 1.3. A Cauchy-Schwarz inequality shows that Z
Rd
u(t, x)−u?(t, x) dx
2
≤ Z
Rd
u?(t, x)dx Z
Rd
u(t, x)−u?(t, x)2 u?(t, x) dx
≤(1 + 2t)−λγ,0 Z
Rd
u0dx Z
Rd
u(0, x)−u?(0, x)2 u?(0, x) dx . The H¨older interpolation inequality
ku(t,·)−u?(t,·)kp≤ ku(t,·)−u?(t,·)k
1 p
1 ku(t,·)−u?(t,·)k1−∞ 1p
combined with the results of Proposition2.4and Corollary2.7concludes the proof after taking (15) and the expression ofλγ,0 stated in Lemma2.6into account.
3. Decay estimate for the kinetic equation with weak confinement. In this section, we prove Theorem 1.4 by revisiting the L2 approach of [13] in the spirit of [7]. However, no mode-by-mode analysis is applicable here due to the confining potential and the standard Nash inequality has to be replaced by a suitable Caffarelli-Kohn-Nirenberg inequality, which requires moment estimates. The main difference with [7] is to rely on the moments, as was already done in the proof of Theorem1.2.
3.1. Notations and elementary computations. On the space L2(M−1dx dv), we define the scalar product
hf, gi= Z Z
Rd×Rd
f g eVM−1dx dv
and the norm kfk = hf, fi1/2. Let Π be the orthogonal projection operator on Ker(L) given by Πf := M ρ[f], where ρ[f] :=R
Rdf(v)dv, and Tbe the transport operator such thatTf =v· ∇xf− ∇xV · ∇vf. We assume that
M(v) = (2π)−d2e−12|v|2 ∀v∈Rd.
Let us use the notationu[f] :=eV ρ[f] and observe that T Πf =M e−V v· ∇xu[f], (T Π)∗f =−M∇x·ρ[v f],
(TΠ)∗(TΠ)f =−M∇x· e−V ∇xu[f] , where the last identity follows from R
RdM(v)v⊗v dv = Id. To build a suitable Lyapunov functional, as in [12,13, 7] we introduce the operatorAdefined by
A:= Id+ (TΠ)∗(TΠ)−1 (TΠ)∗. As in [13] we define the Lyapunov functionalHby
H[f] := 1
2kfk2+εhAf , fi and obtain by a direct computation that
d
dtH[f] =−D[f]
with
D[f] :=− hLf , fi+εhATΠf ,Πfi+εhAT(Id−Π)f ,Πfi
− εhTA(Id−Π)f , fi −εhAL(Id−Π)f ,Πfi, (17) where we have used thathAf ,Lfi= 0 andAΠ= 0. The latter being a consequence of the identity ΠTΠ = 0, that has been called the “diffusive macroscopic limit”
in [13]. For the first term in D[f], we rely on themicroscopic coercivity estimate (see [13])
− hLf , fi ≥λmk(Id−Π)fk2.
The second term hATΠf ,Πfi is expected to control kΠfk2. In Section 3.2, the remaining terms will be estimated to show that for ε small enough D[f] controls k(Id−Π)fk2+hATΠf ,Πfi. As in Section2.2, estimates on moments are needed, which will be established in Section 3.3 and used in Section 3.4 to show a Nash type estimate and to complete the proof of Theorem 1.4 by relating the entropy dissipationD[f] toH[f] and by solving the resulting differential inequality.
3.2. Proof of the Lyapunov functional property of H[f]. Let us define the notations
hu1, u2iV :=
Z
Rd
u1u2e−V dx and kuk2V :=hu, uiV associated with the norm L2(e−V dx). Unless it is specified,∇ means∇x. Lemma 3.1. With the above notations, we have
kAfk ≤ 1
2k(Id−Π)fk, kTAfk ≤ k(Id−Π)fk and
|hTA(Id−Π)f , fi| ≤ k(Id−Π)fk2.
Proof. We already know from [13, Lemma 1] that the operatorTAis bounded. Let us give a short proof for completeness. The equationAf =gis equivalent to
(TΠ)∗f =g+ (TΠ)∗(TΠ)g . (18)
Multiplying (18) byg M−1eV, we get that kgk2+kTΠgk2=hf,TΠgi=h(Id−Π)f ,TΠgi
≤ k(Id−Π)fk kTΠgk ≤ 1
4k(Id−Π)fk2+kTΠgk2 from which we deduce that kAfk = kgk ≤ 12k(Id−Π)fk. Since A = ΠA, be- cause (18) can be rewritten asg=ΠT2Πg−ΠTf using (TΠ)∗=−ΠT, we also have thatTAf =TΠg and obtain thatkTAfk=kTΠgk ≤ k(Id−Π)fk.
By taking into account the expression ofT, Equation (18) amounts tog=Πg= w M e−V wherewsolves
w− Lw+∇x·j= 0 andj=R
Rdv f dx=R
Rdv(Id−Π)f dx. Hence hTA(Id−Π)f , hi=
Z
Rd
(v· ∇xw)h dx=hTA(Id−Π)f ,(Id−Π)hi. This applies tof =h, so that
hTA(Id−Π)f , fi=hTA(Id−Π)f ,(Id−Π)fi ≤ k(Id−Π)fk2.
The term hATΠf ,Πfi is the one which gives the macroscopic decay rate. Let w[f] be such that Id+ (TΠ)∗(TΠ)−1
Πf =w M e−V. Thenwsolves w− Lw=u[f] where Lw:=eV ∇ · e−V ∇w
. (19)
Lemma 3.2. With the above notations, ifu=u[f] and w=w[f] solves (19), we have
hATΠf ,Πfi=k∇wk2V +kLwk2V ≤ 5 4kuk2V . Proof. Letwbe a solution of (19). Since
ATΠf = Id+ (TΠ)∗(TΠ)−1
(TΠ)∗(TΠ)Πf
= Id+ (TΠ)∗(TΠ)−1
Id+ (TΠ)∗TΠ−Id Πf
=Πf− Id+ (TΠ)∗(TΠ)−1
Πf =Πf −w M e−V , we obtain that
ATΠf = (u−w)M e−V .
Using (19) and integrating on Rd after multiplying byΠf =u M e−V, we obtain that
hATΠf ,Πfi=hu, u−wiV =hw− Lw,−LwiV =k∇wk2V +kLwk2V . On the other hand, we can also write that
hATΠf ,Πfi=hu, u−wiV =− hu,LwiV and obtain that
k∇wk2V +kLwk2V =− hu,LwiV ≤ kukV kLwkV ≤1
4kuk2V +kLwk2V , using the Cauchy-Schwarz inequality. As a consequence, we obtain that
k∇wk2V ≤ 1
4kuk2V and kLwkV ≤ kukV , which concludes the proof.
Lemma 3.3. With the above notations, if u=u[f]andw solves (19), we have kHess(w)k2V ≤max{1, γ} hATΠf ,Πfi.
Proof. The operatorL= ∆− ∇V · ∇is such that [L,∇]w=L(∇w)− ∇(Lw) =
L ∂x∂w
i
−∂x∂
i(Lw)d i=1
= Hess(V)· ∇w and it is self-adjoint on L2(eV dx) so that
hLw1, w2iV =− h∇w1,∇w2iV =hw1,Lw2iV
for any w1 and w2. Applied first with w1 = w and w2 = Lw and then with w1=w2=∇w, this shows that
kLwk2V =− h∇w,∇LwiV =− h∇w,L∇wiV + Z
Rd
∇w·[L,∇]w e−V dx
=kHess(w)k2V + Z
Rd
Hess(V) : (∇w⊗ ∇w)e−V dx where kHess(w)k22 = R
Rd|Hess(w)|2e−V dx =Pd i,j=1
R
Rd
∂2w
∂xi∂xj
2
e−Vdx. In the caseV =V2, we deduce from
∂2V
∂xi∂xj = γ hxi2
δij−2xixj
hxi2
that
Hess(V)−γId. Hence
max{1, γ} hATΠf ,Πfi ≥ kLwk2V + max{1, γ} k∇wk2V
≥ kHess(w)k2V −γk∇wk2V + max{1, γ} k∇wk2V , which concludes the proof.
Lemma 3.4. With the above notations and withmγ := 3 max{1, γ}, we have
|hAT(Id−Π)f ,Πfi| ≤mγ hATΠf ,Πfi1/2 k(Id−Π)fk.
Proof. Assume that u=u[f] andw solves (19). Using g = Id+ (TΠ)∗(TΠ)−1
f so that Id+ (TΠ)∗(TΠ)
g=f meansg−(Lw)M e−V =f, let us compute hAT(Id−Π)f ,Πfi=hT(Id−Π)f ,A∗Πfi=hT(Id−Π)f ,TΠgi
=−
(Id−Π)f ,T2Πg
=− Z Z
Rd×Rd
√
M v⊗v(Id−Π)f
√M : Hess(w)dx dv
=− Z Z
Rd×Rd
√
M v⊗v−1dId(Id−Π)f
√
M : Hess(w)dx dv , where we use that T is antisymmetric, and the fact that TΠg = (v· ∇xw)M e−V ifΠg=w M e−V, so thatT2Πg = v⊗v : Hess(w)
M e−V. We conclude using a Cauchy-Schwarz inequality, Lemma3.2and Lemma3.3.
In order to have unified notations, we adopt the convention thatσ= 1/√ 2 if L is the Fokker-Planck operator.
Lemma 3.5. With the above notations, we have hAL(Id−Π)f ,Πfi ≤√
2σhATΠf ,Πfi1/2 k(Id−Π)fk. Proof. We use duality to write
hAL(Id−Π)f ,Πfi=hL(Id−Π)f , hi whereh=A∗f = (TΠ)gand g= Id+ (TΠ)∗(TΠ)−1
f so that Id+ (TΠ)∗(TΠ)
g=f andh= (v· ∇w)M e−V. Herewsolves (19) withu=u[f].
•IfLis the Fokker-Planck operator, then R
RdvLf dv=−j so that
|hAL(Id−Π)f , fi|=|hj,∇wi2| ≤ kjkVk∇wkV .
We know from Lemma3.2thatk∇wkV ≤ hATΠf ,Πfi1/2. The estimate onj=|j|e wheree∈Sd−1, goes as follows: by computing
|j|= Z
Rd
v f dv
= Z
Rd
v(Id−Π)f dv
≤ Z
Rd
(Id−Π)f M−1/2 |v·e|M1/2 dv
≤ Z
Rd
Id−Π)f
2M−1dv Z
Rd
|v·e|2M dv 12
= Z
Rd
Id−Π)f
2M−1dv 12
, we know that
kj eVk2V = Z
Rd
|j|2eV dx≤ Z Z
Rd×Rd
Id−Π)f
2M−1eV dx dv=k(Id−Π)fk2.
•IfLis the scattering operator, then kL(Id−Π)fk2≤σ2
Z
Rd
1 M
Z
Rd
M M0
f0 M0 − f
M
dv0
2
dv
≤σ2 Z
Rd
M Z
Rd
√ M0√
M0
f0 M0 − f
M
dv0
2
dv
≤σ2 Z Z
Rd×Rd
M M0
f0 M0 − f
M
2
dv dv0≤4σ2 Z
Rd
f2M−1dv andkhk=
(v· ∇w)M e−V
=k∇wk2so that hAL(Id−Π)f , fi ≤√
2σhATΠf ,Πfi1/2k(Id−Π)fk.
Notice that for a nonnegative functionf, we have the improved boundskL(Id−Π)k ≤ σk(Id−Π)fk andhAL(Id−Π)f ,Πfi ≤σ hATΠf ,Πfi1/2 k(Id−Π)fk.
Finally, we apply the results of Lemmas3.1,3.4,3.5to the right hand side of (17):
Lemma 3.6. With the above notations, we have
D[f]≥λε k(Id−Π)fk2+hATΠf ,Πfi
with
λε:= 1 2
λm−
q
(λm−2ε)2+ε2 mγ+√ 2σ2
andλε>0, ifε >0 is small enough.
The functional H[f] is a Lyapunov function in the sense thatD[f]≥0 and the equationD[f] = 0 has a unique solutionf = 0.
Proof. The above mentioned Lemmas imply D[f]≥(λm−ε)k(Id−Π)fk2+εhATΠf ,Πfi
−ε
mγ+√ 2σ
k(Id−Π)fk hATΠf ,Πfi1/2 . The Lyapunov function property is a consequence of (19) and Lemma3.2.
3.3. Moment estimates. Let us consider the case V = V2 and define the kth order moments inxandvby
Jk(t) :=khxikf(t,·,·)k1 and Kk(t) :=k|v|kf(t,·,·)k1.
Our goal is to prove estimates onJk andKk. Notice thatJ0=K0=kf0kL1(Rd×Rd)
is constant iff solves (5).
Lemma 3.7. Let d ≥1, γ ∈ (0, d), k ∈ N with k ≥2,V =V2 and assume that f ∈C R+,L2(M−1dx dv)
is a nonnegative solution of (5) with initial datum f0 such thatRR
Rd×Rdhxikf0dx dv <+∞andRR
Rd×Rd|v|kf0dx dv <+∞. There exist constants C2, . . . , Ck such that
J`(t)≤C` (1 +t)`/2 and K`(t)≤C` ∀t≥0, `= 2, . . . , k . (20) Proof. We present the proof for a Fokker-Planck operator, the case of a scattering operator follows the same steps. A direct computation shows that
dK`
dt ≤` γ Z Z
Rd×Rd
|x·v|
hxi2 |v|`−2f(t, x, v)dx dv+`(`+d−2)K`−2−` K`. A boundC` forK`,`∈N, follows after observing that
Z Z
Rd×Rd
|x·v|
hxi2 |v|`−2f(t, x, v)dx dv≤K`−1≤K01/`K`1−1/`
andK`−2≤K02/`K`1−2/` using H¨older’s inequality twice.
Next, let us compute dJ`
dt =` Z Z
Rd×Rd
hxi`−2x·v f(t, x, v)dx dv=:` L`, and
dL`
dt = Z Z
Rd×Rd
hxi`−2|v|2f dx dv+ (`−2) Z Z
Rd×Rd
hxi`−4(x·v)2f dx dv
−γ Z Z
Rd×Rd
hxi`−4|x|2f dx dv−L`
≤(`−1) Z Z
Rd×Rd
hxi`−2|v|2f dx dv−L`. (21) Note that, again by H¨older’s inequality,|L`| ≤J`1−1/`K`1/`,`= 2, . . . , k.
We prove the bound on J`(t) by induction. If ` = 2, (21) implies L2(t) ≤ max
L2(0), C2 and, thus,J2(t)≤C2(1 +t), up to a redefinition ofC2. Now let` >2 and assume that
J`−1(t)≤C`−1(1 +t)`−12 .
We use H¨older’s inequality once more for the right hand side of (21):
dL`
dt ≤(`−1)J
`−2
`−1
`−1K
1
`−1
2(`−1)−L`≤(`−1)C
`−2
`−1
`−1 C
1
`−1
2(`−1)(1 +t)`2−1−L`, which implies
L`≤C (1 +t)`2−1,
and one more integration with respect to testablishes the estimate for J` in (20), up to an eventual redefinition ofC`.
Lemma 3.8. Let d≥1,γ∈(0, d),k∈Nwithk >2,V =V2and assume thatf ∈ C R+,L2(M−1dx dv)
is a nonnegative solution of (5)with initial datum f0 such that RR
Rd×Rdhxikf0dx dv <+∞andRR
Rd×Rd|v|kf0dx dv <+∞. Let w=w[f] be determined by (19) in terms ofu=u[f]. Then there exists a positive constant Ck
such that
0≤Mk(t) :=
Z
Rd
whxik−γdx≤Ck(1 +t)k/2 ∀t≥0.
Proof. The solutionwof (19) is positive by the maximum principle. In what follows we use the definition ofM`for arbitrary integers`and note that for`≤0,
M`≤M0= Z
Rd
w e−V dx= Z
Rd
u e−V dx=kf0k1. (22) Multiplication of (19) byhxi`−γ and integration overRd gives
M`=`(`−2 +d−γ)M`−2−(`−2−γ)M`−4+J`, (23) where J` has been estimated in Lemma3.7. Then, with ` = 2 and (22), we ob- tain M2(t) ≤ C2(1 +t). This implies by the H¨older inequality that M1(t) ≤ pM0M2(t) ≤ C1(1 +t)1/2. For 2 < ` ≤ k the estimate M`(t) ≤ C`(1 +t)`/2 follows recursively from (23).
3.4. Decay estimate for the kinetic equation (proof of Theorem 1.4).
Lemma 3.9. Let d ≥ 1, γ ∈ (0, d), k ≥ max{2, γ/2}, V =V2 and assume that f ∈C R+,L2(M−1dx dv)
is a nonnegative solution of (5) with initial datum f0 such thatRR
Rd×Rdhxikf0dx dv <+∞andRR
Rd×Rd|v|kf0dx dv <+∞. Assume the above notations, in particular withMkdefined as in Lemma3.8, with the constantK from (30)(cf. AppendixB), and witha= d+2+2k−γd+2k−γ . Then
kΠfk2≤2hATΠf ,Πfi+KMk2(1−a)hATΠf ,Πfia=: Φ (hATΠf ,Πfi;Mk) ∀t≥0. Proof. Ifu=u[f] andwsolves (19), we recall that
hATΠf ,Πfi=k∇wk2V +kLwk2V by Lemma3.2. From (19), we also deduce that
kuk2V =hu, w− LwiV ≤ kukV kwk2V + 2k∇wk2V +kLwk2V1/2 . By inequality (31) of AppendixB, we have that
kwk2V ≤ K k∇wk2aV Mk2(1−a)
ifγ≤2 (d−2) (otherwise, one has to use ineq. (30) of AppendixB: details are left to the reader). Combining these inequalities gives
kuk2V ≤ K k∇wk2aV Mk2(1−a)+k∇wk2V +hATΠf ,Πfi, which, noting thatkΠfk=kukV, implies the result.
As a consequence of Lemmas3.1,3.6, 3.9and of the properties of Φ we have H[f] =1
2kfk2+εhAf , fi
≤1 +ε
2 kfk2≤1 +ε
2 k(Id−Π)fk2+ Φ (hATΠf ,Πfi;Mk)
≤1 +ε
2 Φ k(Id−Π)fk2+hATΠf ,Πfi;Mk
≤ 1 +ε
2 Φ
D[f] λε ;Mk
, implying, with Lemma3.8,
dH[f]
dt =−D[f]≤ −λεΦ−1 2
1 +εH[f];Ck(1 +t)k/2
.
The decay of H[f] can be estimated by the solution z of the corresponding ODE problem
dz
dt =−λεΦ−1 2
1 +εz;Ck(1 +t)k/2
, z(0) =H[f0].
By the properties of Φ it is obvious thatz(t)→0 monotonically ast→+∞, which implies that the same is true for dzdt. Therefore, there existst0>0 such that, in the rewritten ODE
−2 λε
dz
dt +KCk2(1−a)(1 +t)k(1−a)
−1 λε
dz dt
a
= 2z 1 +ε,
the first term is smaller than the second fort≥t0, implying the differential inequal- ity
dz
dt ≤ −κ z1/a(1 +t)k(1−1/a) fort≥t0,
with an appropriately defined positive constantκ. Integration and estimation as in Section2.2give
z(t)≤C(1 +t)
1+k(1−1/a)
1−1/a =C(1 +t)γ−d2 , thus completing the proof of Theorem1.4.
Acknowledgments: This work has been partially supported by the Projects EFI ANR-17-CE40-0030 (E.B., J.D.) of the French National Research Agency. The work of C.S. has been supported by the Austrian Science Foundation (grants no.
F65 and W1245), by the Fondation Sciences Math´ematiques de Paris, and by Paris Sciences et Lettres. All authors are part of the Amadeus project Hypocoercivity no. 39453PH.
The authors warmly thank an anonymous referee for his careful reading, which allowed to remove some ambiguities and typos, and suggested some significant im- provements, particularly for dimensionsd= 1 andd= 2.
c 2019 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
