Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations
Tome 4 (2017), p. 389-433.
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UNIFORM SEMIGROUP SPECTRAL ANALYSIS OF THE DISCRETE, FRACTIONAL AND CLASSICAL
FOKKER-PLANCK EQUATIONS
by Stéphane Mischler & Isabelle Tristani
Abstract. — In this paper, we investigate the spectral analysis and long time asymptotic con- vergence of semigroups associated to discrete, fractional and classical Fokker-Planck equations in some regime where the corresponding operators are close. We successively deal with the discrete and the classical Fokker-Planck model, the fractional and the classical Fokker-Planck model and finally the discrete and the fractional Fokker-Planck model. In each case, we prove uniform spectral estimates using perturbation and/or enlargement arguments.
Résumé(Analyse spectrale uniforme des équations de Fokker-Planck discrète, fractionnaire et classique)
Dans cet article, nous nous intéressons à l’analyse spectrale et au comportement asympto- tique en temps long des semi-groupes associés aux équations de Fokker-Planck discrète, frac- tionnaire et classique dans des régimes où les opérateurs correspondants sont proches. Nous traitons successivement les modèles de Fokker-Planck discret et classique, puis fractionnaire et classique et enfin discret et fractionnaire. Dans chaque cas, nous démontrons des estimations spectrales uniformes en utilisant des arguments de perturbation et/ou d’élargissement.
Contents
1. Introduction. . . 390
2. From discrete to classical Fokker-Planck equation . . . 395
3. From fractional to classical Fokker-Planck equation. . . 410
4. From discrete to fractional Fokker-Planck equation. . . 413
References. . . 432
Mathematical subject classification (2010). — 47G20, 35B40, 35Q84.
Keywords. — Fokker-Planck equation, fractional Laplacian, spectral gap, exponential rate of con- vergence, long-time asymptotic, semigroup, dissipativity.
The research leading to this paper was (partially) funded by the French ANR project Stab: ANR- 12-BS01-0019. The second author has been partially supported by the fellowship l’Oréal-UNESCO For Women in Science.
1. Introduction
1.1. Models and main result. — In this paper, we are interested in the spectral analysis and the long time asymptotic convergence of semigroups associated to some discrete, fractional and classical Fokker-Planck equations. They are simple models for describing the time evolution of a density function f =f(t, x), t > 0, x ∈ Rd, of particles undergoing both diffusion and (harmonic) confinement mechanisms and write
(1.1) ∂tf = Λf =Df+ div(xf), f(0) =f0. The diffusion term may either be a discrete diffusion
Df = ∆κf :=κ∗f− kκkL1f,
for a convenient (at least nonnegative and symmetric) kernel κ. It can also be a fractional diffusion
(Df)(x) =−(−∆)α/2f(x) :=cα
Z
Rd
f(y)−f(x)−χ(x−y)(x−y)· ∇f(x)
|x−y|d+α dy, (1.2)
withα∈(0,2),χ∈ D(Rd)radially symmetric satisfying the inequality1B(0,1)6χ6 1B(0,2), and a convenient normalization constantcα>0. It can finally be the classical diffusion
Df = ∆f :=
d
X
i=1
∂2xixif.
The main features of these equations are (expected to be) the same: they are mass preserving, namely
hfti=hf0i, ∀t>0, hfi:=
Z
Rd
f dx,
positivity preserving, have a unique positive stationary state with unit mass that we denote byGhere and that stationary state is exponentially stable, meaning that
(1.3) ft−→ hf0iG as t−→ ∞,
with an exponential rate for any solution ft associated to an initial datum f0 with masshf0i. Such results can be obtained using different tools as the spectral analysis of self-adjoint operators, some (generalization of) Poincaré inequalities or logarithmic Sobolev inequalities as well as the Krein-Rutman theory for positive semigroup.
The aim of this paper is to deal with the above generalized Fokker-Planck equations in an unified way and, more importantly, to establish that the convergence (1.3) is exponentially fast for a large class of initial data taken in a fixed weighted Lebesgue or weighted Sobolev spaceX, with a rate of convergence which can be chosen uniformly with respect to the diffusion term.
We investigate three regimes where these diffusion operators are close and for which such a uniform convergence can be established. In Section 2, we first consider the case
when the diffusion operator is discrete
Df =Dεf := ∆κεf, κε:= 1 ε2kε,
where k is a nonnegative, symmetric, normalized, smooth and decaying fast enough kernel and where we use the notation kε(x) = k(x/ε)/εd, ε >0. In the limitε→0, one then recovers the classical diffusion operatorD0= ∆.
In Section 3, we next consider the case when the diffusion operator is fractional Df =Dεf :=−(−∆)(2−ε)/2f, ε∈(0,2),
so that in the limitε→0we also recover the classical diffusion operatorD0= ∆.
In Section 4, we finally consider the case when the diffusion operator is a discrete version of the fractional diffusion, namely
Df =Dεf := ∆κεf,
where (κε) is a family of convenient bounded kernels which converges towards the kernel of the fractional diffusion operatork0:=cα| · |−d−α for some fixedα∈(0,2), in particular, in the limit ε → 0, one may recover the fractional diffusion operator D0=−(−∆)α/2.
In order to write a rough version of our main result, we introduce some notation. We define the weighted Lebesgue spaceL1r,r>0, as the space of measurable functionsf such that fhxir∈L1, wherehxi2:= 1 +|x|2. For any f0∈L1r, we denote asftthe solution to the generalized Fokker-Planck equation (1.1) with initial datum f0 and then we define the semigroup SΛε onX by settingSΛε(t)f0:=ft.
Theorem1.1(rough version). — There exist r >0 andε0∈(0,2)such that for any ε ∈[0, ε0], the semigroup SΛε is well-defined on X := L1r and there exists a unique positive and normalized stationary solution Gε ∈ X to (1.1). Moreover, there exist a <0 andC>1 such that for anyf0∈X, there holds
(1.4) kSΛε(t)f0−Gεhf0ikX 6C eatkf0−Gεhf0ikX, ∀t>0.
Our approach is a semigroup approach in the spirit of the semigroup decomposition framework formalized by Mouhot in [13] and developed subsequently in [8, 5, 14, 9, 7].
Theorem 1.1 generalizes to the discrete diffusion Fokker-Planck equation and to the discrete fractional diffusion Fokker-Planck equation similar results obtained for the classical Fokker-Planck equation in [5, 9] (Section 2) and for the fractional one in [14]
(Section 4). It also makes uniform with respect to the fractional diffusion parameter the convergence results obtained for the fractional diffusion equation in [14] (Sec- tion 3). It is worth mentioning that there exists a huge literature on the long-time behaviour for the Fokker-Planck equation as well as (to a lesser extent) for the frac- tional Fokker-Planck equation. We refer to the references quoted in [5, 9, 14] for details. There also probably exist many papers on the discrete diffusion equation since it is strongly related to a standard random walk inRd, but we were not able to find any precise reference in this PDE context.
1.2. Method of proof. — Let us explain our approach. First, we may associate a semigroup SΛε to the evolution equation (1.1) in many Sobolev spaces, and such a semigroup is mass preserving and positive. In other words,SΛεis a Markov semigroup and it is then expected that there exists a unique positive and unit mass steady stateGεto the equation (1.1). Next, we are able to establish that the semigroupSΛε
splits as
SΛε=Sε1+Sε2,
Sε1≈etTε, Tεfinite dimensional, Sε2=O(eat), a <0, (1.5)
in these many weighted Sobolev spaces. The above decomposition of the semigroup is the main technical issue of the paper. It is obtained by introducing a convenient splitting
(1.6) Λε=Aε+Bε,
whereBεenjoys suitable dissipativity property andAεenjoys some suitableBε-power regularity (a property that we introduce in Section 2.4 (see also [7]) and that we name in that way by analogy with the Bε-power compactness notion introduced by Voigt [16]). Roughly speaking, we are able to establish that the iterated convolution
(AεSBε)(∗n) enjoys some regularization property for somen>1, where for two functions of timeU andV we define the convolution product
(U∗V)(t) :=
Z t 0
U(t−s)V(s)ds
as well as the iterated convolution product byU(∗0)=I,U(∗m)=U ∗U(∗(m−1)), for anym>1. It is worth emphasizing that we are able to exhibit such a splitting with uniform (dissipativity, regularity) estimates with respect to the diffusion parameter ε∈[0, ε0]in several weighted Sobolev spaces.
As a consequence of (1.5), we may indeed apply the Krein-Rutman theory devel- oped in [11, 7] and exhibit such a unique positive and unit mass steady state Gε. Of course for the classical and fractional Fokker-Planck equations the steady state is trivially given by means of an explicit formula (the Krein-Rutman theory is use- less in that cases). A next direct consequence of the above spectral and semigroup decomposition (1.5) is that there is a spectral gap in the spectral set Σ(Λε) of the generatorΛε, namely
(1.7) λε:= sup{<e ξ∈Σ(Λε)r{0}}<0,
and next that an exponential trend to the equilibrium can be established, namely (1.8) kSΛε(t)f0kX 6Cεeatkf0kX ∀t>0, ∀ε∈[0, ε0], ∀a > λε,
for any initial datumf0∈X with vanishing mass.
Our final step consists in proving that the spectral gap (1.7) and the estimate (1.8) are uniform with respect toε, more precisely, there exists λ∗<0such that λε6λ∗ for anyε∈[0, ε0]and Cε can be chosen independent toε∈[0, ε0].
A first way to get such uniform bounds is just to have in at least one Hilbert space Eε⊂L1(Rd)the estimate
∀f ∈ D(Rd), hfi= 0, (Λεf, f)Eε 6λ∗kfk2Eε.
Estimate (1.8) then essentially follows from the fact that the splitting (1.6) holds with operators which are uniformly bounded with respect to ε ∈ [0, ε0]. It is the strategy we use in the case of the fractional diffusion (Section 3) and the work has already been made in [14] except for the simple but fundamental observation that the fractional diffusion operator is uniformly bounded (and converges to the classical diffusion operator) when it is suitable (re)scaled.
A second way to get the desired uniform estimate is to use a perturbation argument.
Observing that, in the discrete cases (Sections 2 and 4),
∀ε∈[0, ε0], Λε−Λ0=O(ε),
for a suitable operator norm, we are able to deduce that ε 7→ λε is a continuous function at ε = 0, from which we readily conclude. We use here again that the considered models converge to the classical or the fractional Fokker-Planck equation.
In other words, the discrete models can be seen as (singular) perturbations of the limit equations and our analysis takes advantage of such a property in order to capture the asymptotic behaviour of the related spectral objects (spectrum, spectral projector) and to conclude the above uniform spectral decomposition. This kind of perturbative method has been introduced in [8] and improved in [15]. In Section 4, we give a new and improved version of the abstract perturbation argument where some dissipativity assumptions are relaxed with respect to [15] and only required to be satisfied for the limit operator (ε= 0).
1.3. Comments and possible extensions
Motivations. — The main motivation of the present work is rather theoretical and methodological. Spectral gap and semigroup estimates in large Lebesgue spaces have been established both for Boltzmann like equations and Fokker-Planck like equations in a series of recent papers [13, 8, 5, 11, 2, 1, 14, 9, 10]. The proofs are based on a splitting of the generator method as here and previously explained, but the appro- priate splitting are rather different for the two kinds of models. The operatorAε is a multiplication (0-order) operator for a Fokker-Planck equation while it is an integral (−1-order) operator for a Boltzmann equation. More importantly, the fundamental and necessary regularizing effect is given by the action of the semigroupSBε for the Fokker-Planck equation while it is given by the action of the operatorAεfor the Boltz- mann equation. Let us underline here that in Section 4, we exhibit a new splitting for fractional diffusion Fokker-Planck operators (different from the one introduced in [14]) in the spirit of Boltzmann like operators (the operatorAεis an integral oper- ator whereas it was a multiplication operator in [14] and in Section 3). Our purpose is precisely to show that all these equations can be handled in the same framework, by exhibiting a suitable and compatible splitting (1.6) which does not blow up and such
that the time indexed family of operatorsAεSBε (or some iterated convolution prod- ucts of that one) has a good regularizing property which is uniform in the singular limitε→0.
Probability interpretation. — The discrete and fractional Fokker-Planck equations are the evolution equations satisfied by the law of the stochastic process which is solution to the SDE
(1.9) dXt=−Xtdt−dLtε,
where Ltε is the Lévy (jump) process associated to kε/ε2 or cε/|z|d+2−ε. For two trajectories Xt andYtto the above SDE associated to some initial data X0 andY0, andp∈[1,2), we have
d|Xt−Yt|p=−p|Xt−Yt|pdt, from which we deduce
E(|Xt−Yt|p)6e−ptE(|X0−Y0|p), ∀t>0.
We fix nowYt as a stable process for the SDE (1.9). Denoting byfε(t)the law ofXt
andGεthe law ofYt, we classically deduce the Wasserstein distance estimate (1.10) Wp(fε(t), Gε)6e−tWp(f0, Gε), ∀t>0.
In particular, for p = 1, the Kantorovich-Rubinstein Theorem says that (1.10) is equivalent to the estimate
(1.11) kfε(t)−Gεk(W1,∞(Rd))0 6e−tkf0−Gεk(W1,∞(Rd))0, ∀t>0.
Estimates (1.10) and (1.11) have to be compared with (1.8). Proceeding in a similar way as in [11, 9] it is likely that the spectral gap estimate (1.11) can be extended (by “shrinkage of the space”) to a weighted Lebesgue space framework and then to get the estimate in Theorem 1.1 for anya∈(−1,0).
Singular kernel and other confinement term. — We also believe that a similar analysis can be handled with more singular kernels than the ones considered here. The typical example should bek(z) = (δ−1+δ1)/2in dimensiond= 1, and for confinement term different from the harmonic confinement considered here, including other forces or discrete confinement term. In order to perform such an analysis one could use some trick developed in [11] in order to handle the equal mitosis (which uses one more iteration of the convolution product of the time indexed family of operators AεSBε).
Linearized and nonlinear equations. — We also believe that a similar analysis can be adapted to nonlinear equations. The typical example we have in mind is the Landau grazing collision limit of the Boltzmann equation. One can expect to get an exponen- tial trend of solutions to its associated Maxwellian equilibrium which is uniform with respect to the considered model (Boltzmann equation with and without Grad’s cutoff and Landau equation).
Kinetic like models. — A more challenging issue would be to extend the uniform asymptotic analysis to the Langevin SDE or the kinetic Fokker-Planck equation by using some idea developed in [1] which make possible to connect (from a spectral analysis point of view) the parabolic-parabolic Keller-Segel equation to the parabolic- elliptic Keller-Segel equation. The next step should be to apply the theory to the Navier-Stokes diffusion limit of the (in)elastic Boltzmann equation. These more tech- nical problems will be investigated in next works.
1.4. Outline of the paper. — Let us describe the plan of the paper. In each section, we treat a family of equations in a uniform framework, from a spectral analysis view- point with a semigroup approach. In Section 2, we deal with the discrete and classical Fokker-Planck equations. Section 3 is dedicated to the analysis of the fractional and classical Fokker-Planck equations. Finally, Section 4 is devoted to the study of the discrete and fractional Fokker-Planck equations.
1.5. Notation. — For a (measurable) moment functionν :Rd →R+, we define the norms
kfkLp(ν):=kf νkLp(Rd), kfkpWk,p(ν):=
k
X
i=0
k∂ifkpLp(ν), k>1,
and the associated weighted Lebesgue and Sobolev spaces Lp(ν) and Wk,p(ν), we denote Hk(ν) = Wk,2(ν) for k > 1. We also use the shorthand Lpr and Wr1,p for the Lebesgue and Sobolev spacesLp(ν)andW1,p(ν)when the weightν is defined as ν(x) =hxir,hxi:= (1 +|x|2)1/2.
We denote by m a polynomial weight m(x) := hxiq with q > 0, the range of admissibleq will be specified throughout the paper.
In what follows, we will use the same notationC for positive constants that may change from line to line. Moreover, the notation A≈B shall mean that there exist two positive constantsC1,C2 such thatC1A6B6C2A.
2. From discrete to classical Fokker-Planck equation
In this section, we consider a kernel k ∈ W2,1 ∩ L13 which is symmetric, i.e., k(−x) =k(x)for any x∈Rd, satisfies the normalization condition
(2.1)
Z
Rd
k(x)
1 x x⊗x
dx=
1 0 2Id
,
as well as the positivity condition: there existκ0, ρ >0 such that
(2.2) k>κ01B(0,ρ).
We definekε(x) := 1/εdk(x/ε),x∈Rd forε >0, and we consider the discrete and classical Fokker-Planck equations
(2.3)
∂tf = 1
ε2(kε∗f−f) + div(xf) =: Λεf, ε >0,
∂tf = ∆f+ div(xf) =: Λ0f.
The main result of the section reads as follows.
Theorem2.1. — Assume r > d/2 and consider a symmetric kernel k belonging to W2,1∩L12r
0+3 withr0>max(r+d/2,5 +d/2) which satisfies(2.1)and (2.2).
(1) For any ε > 0, there exists a positive and unit mass normalized steady state Gε∈L1r to the discrete Fokker-Planck equation (2.3).
(2) There exist explicit constantsa0 <0 and ε0>0 such that for any ε∈[0, ε0], the semigroupSΛε(t)associated to the discrete Fokker-Planck equation (2.3)satisfies:
for any f0∈L1r and any a > a0, kSΛε(t)f0−Gεhf0ikL1
r6Caeatkf0−Gεhf0ikL1
r, ∀t>0,
for some explicit constant Ca > 1. In particular, the spectrum Σ(Λε) of Λε sat- isfies the separation property Σ(Λε)∩Da0 = {0} in L1r, where we have denoted Dα:={ξ∈Rd; <e ξ > α}.
The method of the proof consists in introducing a suitable splitting of the opera- tor Λε as Λε=Aε+Bε, in establishing some dissipativity and regularity properties on Bε and AεSBε and finally in applying the version [11, 7] of the Krein-Rutman theorem as well as the perturbation theory developed in [8, 15, 7].
2.1. Splitting of Λε. — Let us fix χ ∈ D(Rd) radially symmetric and satisfying 1B(0,1)6χ 61B(0,2). We define χR by χR(x) :=χ(x/R)for R >0 and we denote χcR:= 1−χR.
Forε >0, we define the splittingΛε=Aε+Bε with Aεf :=M χR(kε∗f),
Bεf :=1 ε2 −M
(kε∗f −f) +M χcR(kε∗f−f) + div(xf)−M χRf, for some constants M, R to be chosen later. Similarly, we define the splitting Λ0=A0+B0 with A0f := M χRf and thus B0f := Λ0f −M χRf for some constantsM, R to be chosen later.
2.2. Uniform boundedness ofAε
Lemma2.2. — For anyp∈[1,∞],s>0 and any weight functionν >1, the opera- torAε is bounded fromWs,p intoWs,p(ν)with norm independent of ε.
Proof. — For anyf ∈Lp(ν), we have
kAεfkLp(ν)6Ckkε∗fkLp6CkfkLp.
thanks to the Young’s inequality and because kkεkL1 = kkkL1 = 1. We conclude thatAεis bounded fromLpintoLp(ν). The proof for the cases >0 is similar and it
is thus skipped.
2.3. Uniform dissipativity properties ofBε. — We recall thatm(x) =hxiq. Lemma2.3. — Considerp∈[1,2]andq > d(p−1)/p. Let us suppose thatk∈L1pq+1. For any a > d(1−1/p)−q, there exist ε0>0,M >0 and R>0 such that for any ε∈[0, ε0],Bε−ais dissipative in Lpq, or equivalently
(2.4) h(Bε−a)f,Φ0(f)iLpq 60, ∀f ∈ D(Rd), Φ(f) =|f|p/p.
Proof. — We split the operator in several pieces Bεf =1
ε2 −M
(kε∗f−f) +M χcR(kε∗f−f)
+ div(xf)−M χRf =:Bε1+· · ·+Bε4, and we estimate each term
Ti:=hBiεf,Φ0(f)iLpq = Z
Rd
Bεif
(signf)|f|p−1mpdx
separately. From now on, we consider a > d(1−1/p)−q, we fix ε1 > 0 such that M 61/(2ε21)and we considerε∈(0, ε1].
We first deal withT1. We observe that
(2.5) (f(y)−f(x)) sign(f(x))|f|p−1(x)6 1
p(|f|p(y)− |f|p(x)), using the convexity ofΦ. We then compute
T1=1
ε2 −MZ
Rd×Rd
kε(x−y) (f(y)−f(x)) Φ0(f(x))mp(x)dy dx 6 1
p 1
ε2−MZ
Rd×Rd
(|f|p(y)− |f|p(x))kε(x−y)mp(x)dy dx
= 1 p
1
ε2−MZ
Rd×Rd
(mp(y)−mp(x))kε(x−y)|f|p(x)dy dx,
where we have performed a change of variables to get the last equality. From a Taylor expansion, we have
mp(y)−mp(x) = (y−x)· ∇mp(x) + Θ(x, y), where
|Θ(x, y)|6 1 2
Z 1 0
|D2mp(x+θ(y−x))(y−x, y−x)|dθ 6C|x−y|2hxipq−2hx−yipq−2,
for some constantC∈(0,∞). The term involving the gradient ofmpgives no contri- bution because of (2.1) and we thus obtain
(2.6)
T16C 1−M ε2 Z
Rd×Rd
kε(x−y)|x−y|2
ε2 hx−yipq−2dy|f|p(x)hxipq−2dx 6C
Z
Rd
|f|p(x)hxipq−2dx.
We now treat the second termT2. Proceeding as above and thanks to (2.5) again, we have
T2= Z
Rd×Rd
M χcR(x)kε(x−y) (f(y)−f(x)) Φ0(f(x))mp(x)dy dx 6 M
p Z
Rd×Rd
k(z){χcR(x+εz)mp(x+εz)−χcR(x)mp(x)}dz|f(x)|pdy.
Using the mean value theorem
χcR(x+εz) =χcR(x) +ε z· ∇χcR(x+θεz), mp(x+εz) =mp(x) +εz· ∇mp(x+θ0εz), for someθ, θ0∈(0,1), and the estimates
|∇χcR|6CR and |∇mp(y+θ0εz)|6Chyipq−1hzipq−1, we conclude that
(2.7) T26M CRε
Z
Rd
|f|pmp.
As far asT3 is concerned, we just perform an integration by parts:
(2.8)
T3=d Z
Rd
|f(x)|pmp(x)dx−1 p
Z
Rd
|f(x)|p div(x mp)(x)dx
= Z
Rd
|f(x)|pmp(x) d
1−1 p
−q|x|2 hxi2
dx.
The estimates (2.6), (2.7) and (2.8) together give Z
Rd
BεfΦ0(f)mp6 Z
Rd
|f(x)|pmp(x)
Chxi−2+ d
p0 −q|x|2
hxi2 +M CRε−M χR dx
= Z
Rd
|f|pmp ψR,pε −M χR
,
wherep0=p/(p−1)and we have denoted (2.9) ψR,pε (x) :=Chxi−2+ d
p0 −q|x|2
hxi2 +M CRε.
Because ψεR,p(x)→d/p0−q whenε→0 and |x| → ∞, we can thus chooseM >0, R>0 andε06ε1such that for anyε∈(0, ε0],
∀x∈Rd, ψR,pε (x)6a.
As a conclusion, for such a choice of constants, we obtain (2.4). We refer to [5, 9] for
the proof in the caseε= 0.
Lemma2.4. — Let s∈ N and q > d/2 +s. Assume that k ∈ L12q+1. Then, for any a > d/2−q+s, there exist ε0 >0,M >0 and R>0 such that for anyε∈[0, ε0], Bε−ais hypodissipative inHqs.
Proof. — The cases= 0is nothing but Lemma 2.3 applied withp= 2. We now deal with the cases= 1. We considerft a solution to
∂tft=Bεft, f0=f.
From the previous lemma, we already know that
(2.10) 1
2 d dtkftk2L2
q6 Z
Rd
ft2m2 ψεR,2−M χR . We now want to compute the evolution of the derivative offt:
∂t∂xft=B(∂xft) +M ∂x(χcR) (kε∗ft−ft) +∂xft, which in turn implies that
1 2
d
dtk∂xftk2L2 q =
Z
Rd
(∂xft)∂t(∂xft)m2
= Z
Rd
(∂xft)B(∂xft)m2+ Z
Rd
M ∂x(χcR) (kε∗ft) (∂xft)m2
− Z
Rd
M ∂x(χcR)ft(∂xft)m2+ Z
Rd
(∂xft)2m2
=:T1+T2+T3+T4.
ConcerningT1, using the proof of Lemma 2.3, we obtain
(2.11) T16
Z
Rd
(∂xft)2m2 ψR,2ε −M χR .
Then, to deal withT2, we first notice that using Jensen’s inequality and (2.1), we have
kkε∗fk2L2 q =
Z
Rd
Z
Rd
kε(x−y)f(y)dy 2
m2(x)dx 6
Z
Rd×Rd
kε(x−y)m2(x)dx f2(y)dy
= Z
Rd×Rd
k(z)m2(y+εz)dz f2(y)dy 6C
Z
Rd
k(z)m2(z)dz Z
Rd
f2m2. We thus obtain using thatk∈L12q:
kkε∗fkL2
q 6CkfkL2 q.
The termT2is then treated using the Cauchy-Schwarz inequality, Young’s inequality and the fact that|∂x(χcR)|is bounded by a constant depending only onR:
(2.12)
T26M CRkkε∗xftkL2
qk∂xftkL2 q
6M CRkftkL2
qk∂xftkL2 q
6M CRK(ζ)kftk2L2
q+M CRζk∂xftk2L2 q
for anyζ >0as small as we want.
The termT3is handled using an integration by parts and with the fact that|∂x2(χcR)|
is bounded with a constant which only depends onR:
(2.13) T3=M 2
Z
Rd
∂x2(χcR)ft2m2+M 2
Z
Rd
∂x(χcR)ft2∂x(m2)6M CRkftk2L2 q.
Combining estimates (2.11), (2.12) and (2.13), we easily deduce (2.14) 1
2 d
dtk∂xftk2L2
q 6CR,M,ζ Z
Rd
ft2m2
+ Z
Rd
(∂xft)2m2 ψεR,2+M CRζ+ 1−M χR
.
To conclude the proof in the cases= 1, we introduce the norm
|||f|||2H1
q :=kfk2L2
q+ηk∂xfk2L2
q, η >0.
Combining (2.10) and (2.14), we get (2.15) 1
2 d dt|||ft|||2H1
q 6 Z
Rd
ft2m2 ψR,2ε +η CR,M,ζ−M χR +η
Z
Rd
(∂xft)2m2 ψεR,2+M CRζ+ 1−M χR
.
Using the same strategy as in the proof of Lemma 2.3, ifa > d/2−q+ 1, we can chooseM,Rlarge enough andζ,ε0,η small enough such that we have onRd
ψεR,2+η CR,M,ζ −M χR6a and ψR,2ε +M CRζ+ 1−M χR6a for anyε∈(0, ε0], which implies that
1 2
d
dt|||ft|||2H1
q 6a|||ft|||2H1 q.
The higher order derivatives are treated with the same method introducing a similar
modifiedHqsnorm.
2.4. UniformBε-power regularity ofAε. — In this section we prove that AεSBε and its iterated convolution products fulfill nice regularization and growth estimates.
We introduce the notation (2.16) Iε(f) := 1
2ε2 Z
Rd×Rd
(f(x)−f(y))2kε(x−y)dx dy.
Lemma2.5. — There exists a constantK >0 such that for any ε >0, the following estimate holds:
(2.17) k∇(kε∗f)k2L2 6K Iε(f).
Proof
Step 1. — We prove that the assumptions made onkimply (2.18) |bk(ξ)|26K1−bk(ξ)
|ξ|2 , ∀ξ∈Rd,
for some constantK >0. On the one hand, we havebk(0) = 1,bk(ξ)∈Rbecausek is symmetric andbk∈C0(Rd)becausek∈L1. Moreover, performing a Taylor expansion, using the normalization condition (2.1) and the fact thatk∈L13, we have
bk(ξ) = 1− |ξ|2+O(|ξ|3), ∀ξ∈Rd.
We then deduce that (2.18) holds with K = 1 in a small ball ξ ∈ B(0, δ). On the other hand, for anyξ6= 0, we have
bk(ξ) = Z
Eξ
k(x) cos(ξ·x)dx+ Z
Eξc
k(x) cos(ξ·x)dx
<
Z
Eξ
k(x)dx+ Z
Ecξ
k(x)dx= 1,
where Eξ := {x ∈ Rd ; x·ξ ∈ (0, π), |x| 6 r} so that k(x) cos(ξ·x) < k(x) for any x ∈ Eξ from (2.2). Together with the fact that bk ∈ C0(Rd), we deduce that 1−bk(ξ) > η >0 for any ξ ∈ B(0, δ)c. Last, because k ∈ W1,1, we also have
|ξ|2|bk(ξ)|2 =|∇k(ξ)|c 2 6C for any ξ ∈Rd. We then deduce that (2.18) holds with K=C/η in the setB(0, δ)c.
Step 2. — From the normalization condition (2.1), we have Iε(f) = 1
2ε2 Z
Rd×Rd
f2(x)kε(x−y)dx dy+ 1 2ε2
Z
Rd×Rd
f2(y)kε(x−y)dx dy
− 1 ε2
Z
Rd×Rd
f(x)f(y)kε(x−y)dx dy
= 1 ε2
Z
Rd
f2− Z
Rd
(kε∗f)f
.
As a consequence, using Plancherel formula and the identitykbε(ξ) =bk(ε ξ),∀ξ∈Rd, we get
Iε(f) = 1 ε2
Z
Rd
fb2− Z
Rd
kbεfb2
= Z
Rd
fb2(ξ)1−bk(εξ) ε2 dξ.
Then, we use again Plancherel formula to obtain (2.19) k∂x(kε∗f)k2L2 =kF(∂x(kε∗f))k2L2 =
Z
Rd
|ξ|2bk(εξ)2fb2.
We conclude to (2.17) by using (2.18).
We now introduce the following notationλ:= 1/(2K)>0and go into the analysis of regularization properties of the semigroupAεSBε(t).
Lemma2.6. — Consider s1< s2∈Nandq > d/2 +s2. We suppose that k∈L12q+1. Let M, R andε0 so that the conclusion of Lemma 2.4 holds in both spaces Hqs1 and Hqs2. Then, for any a∈(max{d/2−q+s2,−λ},0), there exists n∈Nsuch that for any ε∈[0, ε0], we have the following estimate
k(AεSBε)(∗n)(t)kB(Hs1
q ,Hqs2)6Caeat, for some constant Ca >0.
Proof. — We first give the proof for the case (s1, s2) = (0,1). We consider a ∈ (max{d/2−q+ 1,−λ},0),α0 andα1such thata > α0> α1>max{d/2−q+ 1,−λ}
andft:=SBε(t)f withf ∈L2q, i.e., that satisfies
∂tft=Bεft, f0=f.
From the proof of Lemma 2.4, there existsε0 such that for anyε∈(0, ε0], we have 1
2 d dtkftk2L2
q 6−1 2
1
ε2 −MZ
Rd×Rd
(f(y)−f(x))2 kε(x−y)m2(x)dy dx+α0kftk2L2 q
6− 1 4ε2
Z
Rd×Rd
(f(y)−f(x))2kε(x−y)dy dx+α0kftk2L2 q
6−1
2Iε(ft) +α0kftk2L2 q
where we have used thatM 61/(2ε2)for anyε∈(0, ε0]. Using Lemma 2.5, we obtain d
dtkftk2L2
q6−2λkkε∗xftk2H˙1+ 2α0kftk2L2 q
62α0kkε∗xftk2H˙1+ 2α0kftk2L2 q. Multiplying this inequality bye−2α0t, it implies that
d dt
kftk2L2
qe−2α0t
62α0kkε∗xftk2H˙1e−2α0t and thus, integrating in time
kftk2L2
qe−2α0t−2α0
Z t 0
kkε∗xfsk2H˙1e−2α0sds6kfk2L2 q. In particular, we obtain
(2.20)
Z ∞ 0
kkε∗xfsk2H˙1e−2α0sds6− 1 2α0
kfk2L2 q. We now want to estimate
Z ∞ 0
kAεSBε(s)fk2H1
qe−2α0sds= Z ∞
0
kAεfsk2L2
qe−2α0sds+ Z ∞
0
k∂x(Aεfs)k2L2
qe−2α0sds 6
Z ∞ 0
kAεfsk2L2
qe−2α0sds+ Z ∞
0
kM ∂x(χR)kε∗xfsk2L2
qe−2α0sds +
Z ∞ 0
kM χR∂x(kε∗xfs)k2L2
qe−2α0sds
=:I1+I2+I3.
Using dissipativity properties ofBεand boundedness ofAε, we get I16
Z ∞ 0
e2α1se−2α0sdskfk2L2
q 6Ckfk2L2 q.
We deal with I2 using the fact that M ∂x(χR) is compactly supported, Young’s in- equality and dissipativity properties ofBε:
I26C Z ∞
0
kkε∗xfsk2L2e−2α0sds6C Z ∞
0
kfsk2L2e−2α0sds
6C Z ∞
0
e2α1se−2α0sdskfk2L2
q 6Ckfk2L2 q. Finally, forI3, we use (2.20) to obtain
I36 Z ∞
0
kkε∗xfsk2H˙1e−2α0sds6Ckfk2L2 q. All together, we have proved
Z ∞ 0
kAεSBε(s)fk2H1
qe−2α0sds6Ckfk2L2 q. Consequently, using the Cauchy-Schwarz inequality, we have
(2.21)
Z ∞ 0
kAεSBε(s)fkH1
qe−asds 2
6 Z ∞
0
kAεSBε(s)fk2H1
qe−2α0sds Z ∞
0
e−2(a−α0)sds 6Ckfk2L2
q.
From the dissipativity of Bε in Hq1 proved in Lemma 2.4 and the fact that Aε is bounded in Hq1, we also have
kAεSBε(s)kH1
q→Hq1e−as6C, ∀s>0.
Using the two last estimates together, we deduce that for any t>0 k(AεSBε)(∗2)(t)fkH1
q 6 Z t
0
kAεSBε(t−s)kH1
q→Hq1kAεSBε(s)fkH1 qds 6C eat
Z ∞ 0
e−askAεSBε(s)fkH1qds 6C eatkfkL2
q. We have thus proved
k(AεSBε)(∗2)(t)kL2q→Hq1 6C eat, which corresponds to the case(s1, s2) = (0,1).
Using the same strategy, we can easily obtain that Z ∞
0
kAεSBε(s)fk2Hs
qe−2asds6Ckfk2Hs−1 q ,
for any s>2, and then conclude the proof of the lemma in the caseε >0. We refer
to [5, 9] for the proof in the caseε= 0.
Lemma2.7. — Considerq > d/2,k∈L12q+1 andM,R,ε0 so that the conclusions of Lemma2.3hold. Then, for anya∈(−q,0), there existsn∈Nsuch that the following estimate holds for anyε∈[0, ε0]:
∀t>0, k(AεSBε)(∗n)(t)kB(L1q,L2q)6Caeat, for some constant Ca >0.
Proof. — We first introduce the formal dual operators ofAεandBε: A∗εφ:=kε∗(M χRφ), B∗εφ:= 1
ε2(kε∗φ−φ)−x· ∇φ−kε∗(M χRφ).
We use the same computation as the one used to deal withT1is the proof of Lemma 2.3 and the Cauchy-Schwarz inequality:
Z
Rd
(B∗εφ)φ6− 1 2ε2
Z
Rd×Rd
kε(x−y) (φ(y)−φ(x))2dy dx + 1
2ε2 Z
Rd×Rd
(φ2(y)−φ2(x))kε(x−y)dy dx +d
2 Z
Rd
φ2+kkε∗(M χRφ)kL2kφkL2. We then notice that the second term equals0 and we use Young’s inequality and the fact thatkkεkL1 = 1to get
Z
Rd
(Bε∗φ)φ6− 1 2ε2
Z
Rd×Rd
kε(x−y) (φ(y)−φ(x))2dy dx +d
2 Z
Rd
φ2+1
2kM χRφk2L2+1 2kφk2L2
6−Iε(φ) +C Z
Rd
φ2
whereIεis defined in (2.16). We also have the following inequality:
Iε(χRφ)6 1 ε2
Z
Rd×Rd
kε(x−y)φ2(x) (χR(y)−χR(x))2dy dx + 1
ε2 Z
Rd×Rd
kε(x−y)χ2R(y) (φ(y)−φ(x))2dy dx 6Ck∇χRk∞
Z
Rd
φ2+ 2Iε(φ).
If we denoteφt:=SBε∗(t)φ, we thus have 1
2 d
dtkφtk2L2 6−λkkε∗(χRφt)k2H˙1+bkφtk2L2, b >0.
Multiplying this inequality bye−bt, we obtain d
dt kφtk2L2e−bt
6−2λkkε∗(χRφt)k2H˙1e−bt, ∀t>0, and integrating in time, we get
(2.22) kφtk2L2e−bt+ 2λ Z t
0
kkε∗(χRφs)k2H˙1e−bsds6kφk2L2
q, ∀t>0.
