On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential
aa r X i v : . [ m a t h . A P ] M a y ON GLOBAL EXISTENCE AND TREND TO THE EQUILIBRIUMFOR THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM WITHEXTERIOR CONFINING POTENTIAL by Fr´ed´eric H´erau & Laurent Thomann
Abstract . —
We prove a global existence result with initial data of low regularity, and prove thetrend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with small non linear termbut with a possibly large exterior confining potential in dimension d = 2 and d = 3. The proof relieson a fixed point argument using sharp estimates (at short and long time scales) of the semi-groupassociated to the Fokker-Planck operator, which were obtained by the first author.
1. Introduction and results1.1. Presentation of the equation. —
Let d = 2 or d = 3. We consider the Vlasov-Poisson-Fokker-Planck system (VPFP for short) with external potential, which reads, for ( t, x, v ) ∈ [0 , + ∞ ) × R d × R d (1.1) ∂ t f + v.∂ x f − ( ε E + ∂ x V e ) .∂ v f − γ∂ v . ( ∂ v + v ) f = 0 ,E ( t, x ) = − | S d − | x | x | d ⋆ x ρ ( t, x ) , where ρ ( t, x ) = Z f ( t, x, v ) dv,f (0 , x, v ) = f ( x, v ) , where x V e ( x ) is a given smooth confining potential (see Assumption 1 below). The constant ε ∈ R is the total charge of the system and in the sequel we assume that either ε > ε < d = 3. The constant γ > γ = 1.The unknown f is the distribution function of the particles. We assume that f ≥ Z f ( x, v ) dxdv = 1, it is then easy to check that once a good existence theory is given, theseproperties are preserved, namely that for all t ≥ f ≥ Z f ( t, x, v ) dxdv = 1 , and we refer to Section 3.1 for more details and other basic results. Mathematics Subject Classification . —
Key words and phrases . —
Vlasov-Poisson-Fokker-Planck equation; non self-adjoint operator; global solutions,return to equilibrium.F.H. is supported by the grant ”NOSEVOL” ANR-2011-BS01019-01. L.T. is supported by the grant “ANA´E”ANR-13-BS01-0010-03.The authors warmly thank Jean Dolbeault for enriching discussions and are grateful to Laurent Di Menza whowas at the origin of this collaboration.
FR´ED´ERIC H´ERAU & LAURENT THOMANN
This equation is a model for a plasma submitted to an external confining electric field (in therepulsive case) and also a model for gravitational systems (in the attractive case). When thereis no external potential ( V e = 0), the equation has been exhaustively studied. First existenceresults were obtained by Victory and O’Dwyer in 2d [ ] and by Rein and Weckler [ ] in 3d forsmall data. Bouchut [ ] showed that the equation is globally well-posed in 3 dimensions usingthe explicit kernel. The long time behavior (without any rate) has been studied with or withoutexternal potential by Bouchut and Dolbeault in [ ], Carillo, Soler and Vazquez [ ], and also byDolbeault in [ ].When there is a confining potential, arbitrary polynomial trend to the equilibrium was es-tablished in [ ] where a first notion of hypocoercivity [ ] was developed and used later tothe full model [ ]. The exponential trend to the equilibrium was shown in the linear case (theFokker-Planck equation) for a general external confining potential in [ ] (see also [ ]). Sofar, in the non-linear case, there is no general result about exponential trend to the equilibrium.In the case of the torus (and V = 0), the strategy of Guo can be applied to many models (seee.g. [
12, 13, 14 ]). In the case when the potential is explicitly given by V e ( x ) = C | x | , a recentresult with small data is given in [ ], following the micro-macro strategy of Guo.In all previous cases (torus, V e = 0 or polynomial of order 2), mention that one can computeexplicitly the Green function of the Fokker-Planck operator and also that exact computationscan be done thanks to vanishing commutators. Here instead we will rely on estimates (in shortand long time) of the linear solution of the Fokker-Planck operator obtained by the first authorin [ ], and our approach allows us to deal with a large class of confining potentials V e . Indeed,in [ , Theorem 1.3 ] a first exponential trend to the equilibrium result for a VPFP type modelwas given, but only for a mollified non-linearity. We will prove here a global existence result inthe full VPFP case, with trend to equilibrium assuming that the initial condition f is localisedand has some Sobolev regularity and under the assumption that the electric field is perturbativein the sense that | ε | ≪ V e and first assume the following Assumption 1 . —
The potential x V e ( x ) satisfies e − V e ∈ S ( R d ) , with V e ≥ and V ′′ e ∈ W ∞ , ∞ ( R d ) . Observe that the assumption V e ≥ V e is bounded frombelow and adding to it a sufficiently large constant.We now introduce the Maxwellian of the equation (1.1)(1.2) M ∞ ( x, v ) = e − ( v / V e ( x )+ ε U ∞ ( x )) R e − ( v / V e ( x )+ ε U ∞ ( x )) dxdv , where U ∞ is a solution of the following Poisson-Emden type equation(1.3) − ∆ U ∞ = e − ( V e + ε U ∞ ) R e − ( V e ( x )+ ε U ∞ ( x )) dx . Actually, one gets that under Assumption 1 and | ε | small enough (assuming additionally that ε > d = 2), the equation (1.3) has a unique (Green) solution U ∞ which belongsto W ∞ , ∞ ( R d ) uniformly w.r.t | ε | (see Propositions 3.5 and 3.6 following results from [ ]). TheMaxwellian M ∞ is then in S ( R dx × R dv ) and is the unique L -normalised steady solution ofequation (1.1). N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK In the case d = 2 and ε <
0, existence and uniqueness of solutions to (1.3) are unclear, that’swhy we do not consider this case.For convenience, we now introduce the effective potential at infinity(1.4) V ∞ def = V e + ε U ∞ so that M ∞ ( x, v ) = e − ( v / V ∞ ( x )) R e − ( v / V ∞ ( x )) dxdv . The second assumption on V e is the following Assumption 2 . —
The so-called Witten operator W = − ∆ x + | ∂ x V e | / − ∆ x V e / has a spectralgap in L ( R d ) . We denote by κ > the minimum of this spectral gap and d/ . Example 1.1 . — As an example, we can check that if V e satisfies Assumption 1 and is suchthat | ∂ x V e ( x ) | −→ | x |−→∞ + ∞ then it satisfies also Assumption 2 since it has a compact resolvent.We introduce now the functional framework on which our analysis is done. We consider theweighted space B built from the standard L space after conjugation with a half power of theMaxwellian(1.5) B def = M / ∞ L = (cid:8) f ∈ S ′ ( R d ) s.t. f / M ∞ ∈ L ( M ∞ dxdv ) (cid:9) . We define the natural scalar product h f, g i = Z f g M − ∞ dxdv, and the corresponding norm k f k B = h f, f i = Z f M − ∞ dxdv. Next, consider the Fokker-Planck operator associated to the potential V ∞ defined by(1.6) K ∞ = v.∂ x − ∂ x V ∞ ( x ) .∂ v − γ∂ v . ( ∂ v + v ) . The last object we need before writing our equation in a suitable way is the limit electric field E ∞ ( x ) = ∂ x U ∞ ( x ) = − | S d − | x | x | d ⋆ x Z M ∞ ( x, v ) dv. With all the previous notations, the VPFP equation (1.1) can be rewritten(1.7) ∂ t f + K ∞ f = ε ( E − E ∞ ) ∂ v f,E ( t, x ) = − | S d − | x | x | d ⋆ x ρ ( t, x ) , where ρ ( t, x ) = Z f ( t, x, v ) dv,f (0 , x, v ) = f ( x, v ) . We define the operator Λ x = − ∂ x . (cid:0) ∂ x + ∂ x V ∞ (cid:1) + 1which is up to a conjugation with M / ∞ the Witten operator introduced in Assumption 2 butdefined on B , and Λ v = − ∂ v . ( ∂ v + v ) + 1 , which is again up to a conjugation the harmonic oscillator in velocity. They both are non-negative selfadjoint unbounded operators in B . We also introduceΛ = − ∂ x . (cid:0) ∂ x + ∂ x V ∞ (cid:1) − ∂ v . ( ∂ v + v ) + 1 = Λ x + Λ v − . FR´ED´ERIC H´ERAU & LAURENT THOMANN
It is clear that 1 ≤ Λ x , Λ v ≤ Λ . As we mentioned previously, if V e satisfies Assumptions 1 and 2, then V ∞ = V e + ε U ∞ alsodoes, and we check in Subsection 3.3 that the operator − ∂ x . (cid:0) ∂ x + ∂ x V ∞ (cid:1) − ∂ v . ( ∂ v + v ) = Λ − B which is, uniformly w.r.t | ε | small,bounded from below by κ / α, β ≥ B α,β = B α,βx,v ( R d ) = (cid:8) f ∈ B : Λ αx f ∈ B and Λ βv f ∈ B (cid:9) , and we endow this space by the norm k f k B α,β = k Λ αx f k B + k Λ βv f k B . In the case α = β we simply define B α = B α,αx,v ( R d ) = (cid:8) f ∈ B : Λ α f ∈ B (cid:9) , with the norm k f k B α = k Λ α f k B ∼ k f k B α,α . We observe that M ∞ ∈ B α,β for all α, β ≥
0, since we have M ∞ ∈ S ( R d ). We are now able to state our global well-posedness results.
Theorem 1.2 . —
Let d = 2 and let f ∈ B ( R ) . Assume moreover that Assumptions 1 and 2are satisfied. Then if ε > is small enough, there exists a unique global mild solution f to (1.1) in the class f ∈ C (cid:0) [0 , + ∞ [ ; B ( R ) (cid:1) . Moreover, the following convergence to equilibrium holds true k f ( t ) − M ∞ k B ≤ C e − κ t/c , ∀ t ≥ , and k E ( t ) − E ∞ k L ∞ ( R ) ≤ C e − κ t/c , ∀ t ≥ . By mild, we mean f and E which satisfy the integral formulation of (1.7), namely(1.9) f ( t ) = e − K ∞ f + ε Z t e − ( t − s ) K ∞ ( E ( s ) − E ∞ ) ∂ v f ( s ) ds,E ( t ) = − | S d − | x | x | d ⋆ x Z f ( t ) dv. In the case d = 3, we need to assume more regularity on the initial condition, but theknown results about the uniqueness of the Poisson-Emden equation (see Subsection 3.2) allowto consider also the case ε < U = 14 π | x | ⋆ x Z f dv, which is such that ∆ U = Z f dv . Then N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK Theorem 1.3 . —
Let d = 3 and / < a < / . Assume that f ∈ B a,a ( R ) ∩ L ∞ ( R ) is suchthat U ∈ W , ∞ ( R ) . Assume moreover that Assumptions 1 and 2 are satisfied. Then if | ε | issmall enough, there exists a unique global mild solution f to (1.1) in the class f ∈ C (cid:0) [0 , + ∞ [ ; B a,a ( R ) (cid:1) ∩ L ∞ loc (cid:0) [0 , + ∞ [ ; L ∞ ( R ) (cid:1) . Moreover, for all a ≤ α < / and a ≤ β < such that α − < β < f ∈ C (cid:0) ]0 , + ∞ [ ; B α,β ( R ) (cid:1) , and the following convergence to equilibrium holds true k f ( t ) − M ∞ k B α,β ≤ C e − κ t/c , ∀ t ≥ , and k E ( t ) − E ∞ k L ∞ ( R ) ≤ C e − κ t/c , ∀ t ≥ . In the previous lines, the constants c, C , C > k V ∞ k W , ∞ where V ∞ wasdefined in (1.4), on k U k W , ∞ and on f .Notice that in Theorem 1.3, the parameters ( α, β ) can be chosen independently from a . Itis likely that the assumption a < / β < U is technical. It is needed here in order to guaranteethat the linearised equation near t = 0 enjoys reasonable spectral estimates. Observe (seeRemark 3.17 for more details), that the assumption f ∈ B a,a ( R ) ∩ L ∞ ( R ) alone ensures that U ∈ W ,p ( R ) for any 2 ≤ p < + ∞ .An analogue of the regularizing estimate (1.11) can also be obtained in Theorem 1.2. Thiscan be proven by getting estimates in some spaces B α,βx,v as in the proof of Theorem 1.3 (seeSection 5). We did not include it here in order to simplify the argument.The proof uses estimates of e − tK ∞ in the space B , obtained in [ ] by the first author.Theorem 1.3 extends [ , Theorem 1.3] where he considered a regularised version of the electricfield E in (1.1), which was so that E ( t ) ∈ L ∞ ( R ) for any f ∈ B . Here we tackle this difficultyby using the Sobolev regularity of f and a gain given by the integration in time. The proofrelies on a fixed point argument in a space based on B α,β in the ( x, v ) variables, and allowingan exponential decay in time.As a consequence of Theorems 1.2 and 1.3, we directly obtain the exponential decay of therelative entropy. Let us define H ( f ( t ) , M ∞ ) = Z Z f ( t ) ln (cid:16) f ( t ) M ∞ (cid:17) dxdv, then Corollary 1.4 . —
Let d = 2 or d = 3 . Then under the assumptions of Theorem 1.2 or Theo-rem 1.3, the corresponding solution f of (1.1) satisfies ≤ H ( f ( t ) , M ∞ ) ≤ Ce − κt/c , where C, c > only depend on second order derivatives of V e + ε U ∞ and on f . We refer to [ , Corollary 1.4] for the proof of this result. FR´ED´ERIC H´ERAU & LAURENT THOMANN
Notations . —
In this paper c, C > denote constants the value of which may change from lineto line. These constants will always be universal, or uniformly bounded with respect to the otherparameters. The rest of the paper is organised as follows. In Section 2 we prove some linear estimates one − tK (where K is a generic linear Fokker-Planck operator). In Section 3 we gather some estimateson solutions of (1.1). Finally, Sections 4 and 5 are devoted to the proofs of Theorems 1.2 and 1.3with fixed points arguments.
2. Semi-group estimates
In this section, we denote by V a generic potential satisfying Assumptions 1 and 2. We alsodenote by K the associated generic linear Fokker-Planck operator K = v.∂ x f − ∂ x V.∂ v − ∂ v . ( ∂ v + v ) . Similarly, the operators Λ x = − ∂ x ( ∂ x + ∂ x V ) + 1, Λ = Λ x + Λ v −
1, the normalized Maxwellian M ( x, v ) = e − ( V ( x )+ v / and spaces of type B α,β are built with respect to this generic poten-tial V . For convenience, we also denote by X = v.∂ x f − ∂ x V.∂ v .The aim of this section is to state some estimates of e − tK in B − type norms. These areconsequences of [ ]. In all the following we pose κ = κ /C , where κ is the spectral gap of the operator W defined in Assumption 2 (with V as a poten-tial), and C is a large constant depending only on derivatives of V ′′ explicitly given in [ ,Theorem 0.1].The operator K is maximal accretive in B (see e.g. [ , Theorem 5.5]). This enables us todefine e − tK and to prove that(2.1) (cid:13)(cid:13) e − tK (cid:13)(cid:13) B → B ≤ . Following [ , Theorem 3.1], operator e − tK −→ Id when t −→
0, strongly in B a,a for any a ≥
0. Observe that all the estimates in this section are independent of the dimension d . Fora complete analysis of the linear Fokker-Planck operator we refer to [ ] or [ ]. We now givesome regularizing estimates for the semi-group associated to K , in the spirit of [ , Section 3]. Proposition 2.1 . —
There exists
C > so that for all α, β ∈ [0 , and all t > k Λ αx e − tK k B → B ≤ C (1 + t − α/ ) , k e − tK Λ αx k B → B ≤ C (1 + t − α/ ) and (2.3) k Λ βv e − tK k B → B ≤ C (1 + t − β/ ) , k e − tK Λ βv k B → B ≤ C (1 + t − β/ ) . In the previous bounds, the constant C only depends on a finite number of derivatives of V . Remark 2.2 . — Note that the exponents 1 / / α = 1 are optimalat least in the case V = 0 and in the case when V is a definite quadratic form in x . This canbe checked since in these both cases, the Green kernel of e − tK is explicit. In the case V = 0we refer to [ ], and when V is quadratic, we refer to the general Mehler formula given in [ ,Section 4 ]. N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK Proof of Proposition 2.1 . — We first prove the estimate (2.3). In [ , Proposition 3.1], reinter-preted in our framework, reads(2.4) (cid:13)(cid:13) ( ∂ v + v ) e − tK (cid:13)(cid:13) B → B ≤ C (1 + t − / ) . For f a solution of the equation ∂ t f + Kf = 0 , f ( t = 0) = f , with normalized initial condition f ∈ C ∞ , and using the regularization property of e − tK , wehave for t > k Λ v f ( t ) k B → B = h Λ v f ( t ) , f ( t ) i = k ( ∂ v + v ) f ( t ) k B → B + k f ( t ) k B → B ≤ C (1 + t − / ) + 1 ≤ C ′ (1 + t − / ) . Using that B , x,v = B and (2.1), we therefore have that (cid:13)(cid:13) e − tK (cid:13)(cid:13) B → B , x,v ≤ C ′ (1 + t − / ) , (cid:13)(cid:13) e − tK (cid:13)(cid:13) B → B , x,v ≤ C ′′ and by interpolation we get that for all 0 ≤ β ≤ (cid:13)(cid:13) e − tK (cid:13)(cid:13) B → B ,βx,v ≤ C (1 + t − / ) β ≤ C b (1 + t − β/ )which reads k Λ βv e − tK k B → B ≤ C β (1 + t − β/ )which is the first result. For the converse estimate, we use that K ∗ , the adjoint of K in B givenby K ∗ = − X − ∂ v . ( ∂ v + v ), has the same properties as K so that for all t > k Λ βv e − tK ∗ k B → B ≤ C ′ β (1 + t − β/ ) . Taking the adjoints of this yields k e − tK Λ βv k B → B ≤ C ′ β (1 + t − β/ ) . Concerning the estimates involving Λ x , the proof is exactly the same as the preceding onewith Λ v replaced by Λ x , β replaced by 3 α , − ∂ v . ( ∂ v + v ) replaced by − ∂ x . ( ∂ x + ∂ x V ( x )) andusing the result from [ , Proposition 3.1] (cid:13)(cid:13) ( ∂ x + ∂ x V ( x )) e − tK (cid:13)(cid:13) B → B ≤ C (1 + t − / ) , instead of (2.4). This concludes the proof.From Proposition 2.1, it is easy to deduce the following Corollary 2.3 . —
Let α, β ∈ [0 , . Then k Λ αx e − ( t − s ) K Λ − βv k B → B ≤ C (cid:0) ( t − s ) − / β/ − α/ + 1 (cid:1) , and k Λ βv e − ( t − s ) K Λ − βv k B → B ≤ C (cid:0) ( t − s ) − / + 1 (cid:1) . Proof . — We only prove the first statement, the second is similar. By (2.7), (2.8) and also usingRemark 2.5 we have k Λ αx e − ( t − s ) K Λ − βv k B → B ≤ k Λ α e − ( t − s ) K Λ − βv k B → B ≤ k Λ α e − ( t − s ) K/ k B k e − ( t − s ) K/ Λ − βv k B ≤ C (cid:0) ( t − s ) − / β/ − α/ + 1 (cid:1) , which was the claim. FR´ED´ERIC H´ERAU & LAURENT THOMANN
We define B ⊥ = n f ∈ B s.t. h f, M ∞ i = Z f dxdv = 0 o the orthogonal of M ∞ in B . At this stage we observe that for f ∈ B α ∩ B ⊥ (2.5) Λ αx f ∈ B ⊥ , Λ αv f ∈ B ⊥ and that for all f ∈ B (2.6) ∂ v f ∈ B ⊥ . For (2.5) we use that the operator Λ x is self-adjoint: h Λ αx f, Mi = h f, Λ αx Mi = 0 since Λ x M = M .The same proof holds for Λ v . The justification of (2.6) is similar using that ∂ ∗ v = − ( v + ∂ v )and ( v + ∂ v ) M = 0.A careful analysis shows that we have in fact the following better results when we restrictto B ⊥ . Proposition 2.4 . —
For all α, β ∈ [0 , there exist C α , C β > so that for all t > k Λ αx e − tK k B ⊥ → B ⊥ ≤ C α (1 + t − α/ ) e − κt , k e − tK Λ αx k B ⊥ → B ⊥ ≤ C α (1 + t − α/ ) e − κt and (2.8) k Λ βv e − tK k B ⊥ → B ⊥ ≤ C β (1 + t − β/ ) e − κt , k e − tK Λ βv k B ⊥ → B ⊥ ≤ C β (1 + t − β/ ) e − κt . In the previous bounds, the constants C α and C β only depend on a finite number of derivativesof V .Proof . — For 0 ≤ t ≤
1, this is a direct consequence of the preceding proof and the fact that B ⊥ is stable by X , Λ x and Λ v and therefore Λ , K and K ∗ by direct computations. For t ≥
1, theproposition is a consequence of the regularizing properties of e − tK stated in [ , Theorem 0.1]and the spectral gap for K : it is proven there that for all s ∈ R , there exist N s > C s > ∀ t > , (cid:13)(cid:13) Λ s e − tK Λ s (cid:13)(cid:13) B ⊥ → B ⊥ ≤ C s ( t s + t − s ) e − κt . Using this and possibly replacing κ by κ/ t ≥
1. This completes theproof.
Remark 2.5 . — In fact possibly replacing once more κ by κ/
2, we also get directly that Propo-sition 2.4 is also true with K replaced by K/
2. We shall use this just below.Similarly to Corollary 2.3 we have the following
Corollary 2.6 . —
Let α, β ∈ [0 , . Then (2.9) k Λ αx e − ( t − s ) K Λ − βv k B ⊥ → B ⊥ ≤ C (cid:0) ( t − s ) − / β/ − α/ + 1 (cid:1) e − κ ( t − s ) , and (2.10) k Λ βv e − ( t − s ) K Λ − βv k B ⊥ → B ⊥ ≤ C (cid:0) ( t − s ) − / + 1 (cid:1) e − κ ( t − s ) . Proposition 2.7 . —
There exists
C > so that for all γ ∈ [0 , and all t ≥ k Λ γ e − tK Λ − γ k B → B ≤ C, and (2.12) k Λ γ e − tK Λ − γ k B ⊥ → B ⊥ ≤ Ce − κt . In the previous bounds, the constant only depends on a finite number of derivatives of V . N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK Proof . — We only give the proof of (2.11), since (2.12) can be obtained with the same argument.Recall the definition (1.8) of the space B α,βx,v . We first note that it is equivalent to show that e − tK is bounded from B γ,γx,v into itself. We first begin with the case γ = 2. We therefore look, for aninitial data f ∈ B , x,v at the equation satisfied by g = Λ f in B . Let us define the operator X = v.∂ x − ∂ x V.∂ v . Since ∂ t f + X f − ∂ v . ( ∂ v + v ) f = 0 , f t =0 = f and from the regularising properties of e − tK , we get ∂ t g + X g − ∂ v . ( ∂ v + v ) g = [ X , Λ ]Λ − g, g t =0 = g where we also used that − ∂ v . ( ∂ v + v ) and Λ commute. Integrating against g in B gives ∂ t k g k ≤ (cid:0) [ X , Λ ]Λ − g, g (cid:1) , since X is skew adjoint and − ∂ v . ( ∂ v + v ) is non-negative. Let us study the right-hand sidecommutator. We have [ X , Λ v ]Λ − = [ v.∂ x − ∂ x V ( x ) .∂ v , Λ v ]Λ − = (cid:0) [ v, Λ v ] ∂ x − [ ∂ v , Λ v ] ∂ x V ( x ) (cid:1) Λ − . This gives with a direct computation (cid:13)(cid:13) [ X , Λ v ]Λ − g (cid:13)(cid:13) B ≤ C k g k B . We can do exactly the same with Λ x (using that V (3) is bounded) and we get on the whole that (cid:13)(cid:13) [ X , Λ ]Λ − g (cid:13)(cid:13) B ≤ C k g k B so that with a new constant C > ∂ t k g k B ≤ C k g k B . We therefore get k g ( t ) k B ≤ e Ct k g k B which we will use for t ∈ [0 , e − tK ([ , Theorem 0.1]), wealso know that for all t ≥ k g ( t ) k B ≤ k f ( t ) k B ≤ C ′ k f k B ≤ C ′ k g k B . Putting these results together give for all t ≥ k g ( t ) k B ≤ C k g k B and therefore e − tK is (uniformly in t >
0) bounded from B to B . Now the result is also clearfor γ = 0 by the semi-group property, and by interpolation we get that e − tK is (uniformly in t >
0) bounded from B γ to B γ . As a conclusion we get k Λ γ e − tK Λ − γ k B → B ≤ C γ , which was the claim.We are now able to state the following interpolation results Lemma 2.8 . —
Let β ∈ [0 , . Then there exists C > so that for all a ∈ [0 , β ] k Λ βv e − tK k B a → B ≤ C (1 + t − ( β − a ) / ) . FR´ED´ERIC H´ERAU & LAURENT THOMANN
Proof . — For a = 0, this follows from Proposition 2.1. Next, set a = β , then for f ∈ B β k Λ βv e − tK f k B ≤ k Λ β e − tK f k B ≤ k Λ β e − tK Λ − β k B → B k Λ β f k B ≤ C k f k B β , by (2.11). The general case a ∈ [0 , β ] is obtained by interpolation. Lemma 2.9 . —
Let ≤ a ≤ . Then for all a ≤ a ≤ a + 2 there exists C > so that for all ≤ t ≤ k Λ a (cid:0) e − tK − k B a → B ≤ Ct ( a − a ) / . Proof . — For a = a , the result follows from (2.11). Now we prove the bound for a = a + 2,and the general result will follow by interpolation. We write (cid:0) − e − tK (cid:1) f = Z t e − sK Kf ds.
Then we use that K : B −→ B is bounded, and by (2.11) we get for all f ∈ B a +2 k Λ a (cid:0) e − tK − k B a → B ≤ Z t k Λ a e − sK Λ − a kk Λ a Kf k ds ≤ Ct k f k B a , hence the result.We conclude this section with a technical result. Lemma 2.10 . —
For all δ ∈ R there exists C δ > so that (2.13) k Λ − δv Λ − v ∂ v Λ δv k B → B ⊥ ≤ C δ . In the previous bound, the constant only depends on a finite number of derivatives of V .Proof . — From to [ , Proposition A.7] we directly get that operator Λ − δv Λ − v ∂ v Λ δv is boundedfrom B to B . Indeed in the symbolic estimates and pseudo-differential scales introduced there,the operator ∂ v is of order 1 with respect to the velocity variable. Now using the stability of B ⊥ by Λ v and (2.6) yield the result. Remark 2.11 . — We shall see in the next section (Section 3.6) that most of the results of thissection remain true when V is perturbed by a less regular term e V ∈ W , ∞ . We will need thisfor the small time analysis of the equation (1.1).
3. Intermediate results
In this section, we gather some intermediate results about the Vlasov-(Poisson)-Fokker-Planckequation. In the first subsection we state some a priori basic properties satisfied by solutions ofthe Fokker-Planck equation and then equation (1.1). In the second one we study more carefullythe Poisson term, and in the last one we recall some facts about the equilibrium state.
N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK In this section, we just recall from [ , Ap-pendix A] some standard and basic results about the behaviour of the solutions of the followinglinear Krammers-Fokker-Planck equation(3.1) ( ∂ t f + v.∂ x f − ( ∂ x V − v ) .∂ v f − f − ε E ( t, x ) ∂ v f − ∆ v f = F,f (0 , x, v ) = f ( x, v ) . Note that equation (1.1) with given field E and V = V e enters in this setting and that the linearFokker-Planck equation corresponds to E = 0. In both cases we take F = 0 and point out thatwe used the commutation estimate − ∂ v ( ∂ v + v ) f = ( ∂ v + v )( − ∂ v ) f − f .For the following, we take T > X = L ([0 , T ] × R dx , H v ( R d )) andconsider the space Y = (cid:8) f ∈ X, ( ∂ t + v.∂ x − ( ∂ x V − v ) .∂ v ) f ∈ X ′ (cid:9) . The following result isclassical and we refer to [ , Appendix A] for the proof. Proposition 3.1 . —
Suppose E ∈ L ∞ ([0 , T ] × R d ) , f ∈ L ( R d ) and F ∈ L ([0 , T ] × R dx , H − v ) . Then there exists a unique weak solution f of the equation (3.1) in the class Y . Moreover ( i ) If f ≥ then f ≥ . ( ii ) If f ∈ L ∞ ( R d ) , then for all ≤ t ≤ T , k f ( t ) k L ∞ ( R d ) ≤ e dt k f k L ∞ ( R d ) . This immediately implies the following a priori estimate on the full problem (1.1).
Corollary 3.2 . —
Let f ∈ L ∞ ( R d ) ∩ L ( R d ) be such that f ≥ and consider a solutionof (1.1) such that the field E ∈ L ∞ ([0 , T ] × R d ) . Then, for all ≤ t ≤ T , f ( t, . ) ≥ and k f ( t ) k L ∞ ( R d ) ≤ e dt k f k L ∞ ( R d ) . The aim of this subsection is toprove that the potential U ∞ associated to the stationary solutions of the Vlasov-Poisson-Fokker-Planck equation is in W ∞ , ∞ ( R d ). Recall that the equation satisfied by U ∞ is(3.2) − ∆ U ∞ = e − ( V e + ε U ∞ ) R e − ( V e + ε U ∞ ) dx where we recall that ε is varying in a small fixed neighbourhood of 0, and that ε > d = 2. d = 3. — When we are in the repulsive interaction case ( ε > ](see also [ ]) under a light hypothesis on the external potential. We first quote his result indimension d = 3 and in the Coulombian case Proposition 3.3 ( [ ] , Section 2) . — Let U e ∈ L ∞ loc ( R ) and M > . Assume that e − U e ∈ L ( R ) ,then there exists a unique solution U ∈ L , ∞ ( R ) of the Poisson-Emden equation (3.3) − ∆ U = M e − ( U e + U ) R e − ( U e + U ) dx . Moreover U ≥ . The main property of U which will be needed in the following is U ≥
0, that’s why we do noteven define precisely the space L , ∞ ( R ). For more details, we address to [ ].We then state another result of Bouchut and Dolbeault in the Newtonian case ( ε < M . FR´ED´ERIC H´ERAU & LAURENT THOMANN
Proposition 3.4 ( [ , Theorem 3.2 and Proposition 3.4] ) . — Assume that e − U e ∈ L ( R ) ∩ L ∞ ( R ) and is not identically equal to . Then there exists M < such that for all M < M ≤ thereexists a bounded continuous function of equation (3.3) such that lim x →∞ U ( x ) = 0 . Now Assumption 1 on the exterior potential V e implies that e − V e ∈ L ( R ) ∩ L ∞ ( R ). As aconsequence we can apply Proposition 3.3 at least in the case when ε is small to U = ε U ∞ , U e = V e , M = ε and d = 3 to (3.2) and we get a unique solution U ∞ in L , ∞ when ε > ε < U ∞ ∈ L ∞ . Notice that in ourcontext, | ε | is small and hence both Propositions 3.3 and 3.4 apply here.Actually, the regularity of U ∞ is improved under the assumption e − V e ∈ S ( R ), and we canalso get some uniformity with respect to the parameter ε . Proposition 3.5 . —
Let d = 3 . Suppose that V e satisfies Assumption 1. Then the uniquesolution U ∞ of the Poisson-Emden equation (3.2) is in W ∞ , ∞ ( R ) , with semi-norms uniformlybounded w.r.t. ε varying in a small fixed neighbourhood of .Proof of Proposition 3.5 . — In order to prove that U ∞ ∈ W ∞ , ∞ , it is sufficient to prove thatthe (Green) solution U ∞ of the following Poisson-Emden-type equation(3.4) − ∆ U ∞ = C − e − ( V e + ε U ∞ ) is in W ∞ , ∞ , where C = Z e − ( V e ( x )+ ε U ∞ ( x )) dx is the normalization constant. We first work on ε U ∞ and note that it is given by ε U ∞ = ε C − π | x | ⋆ e − ( V e + ε U ∞ ) . We then consider the Green solution U e of − ∆ U e = e − V e given by U e = 14 π | x | ⋆ e − V e . From Propositions 3.3 and 3.4 we get directly that ε U ∞ exists, at least for ε varying in a smallneighbourhood of 0, and that it is either non-negative or uniformly bounded. It implies thatthere exists a constant C > ε such that 0 ≤ U ∞ ≤ CU e since we also have U ∞ = C − π | x | ⋆ e − ( V e + ε U ∞ ) . From the Hardy-Littlewood Sobolev inequalities or by a direct computation, we have U e ∈ L p for 3 < p ≤ ∞ . Therefore this is also the case for U ∞ . Since we directly have that − ∆ U ∞ ∈ L p for all p ∈ [1 , ∞ ] from (3.4), we get that − ∆ U ∞ + U ∞ ∈ L p , < p < ∞ and this gives U ∞ ∈ W ,p by elliptic regularity in R d (see for example [ ], [ ]).Now we shall use a bootstrap argument to prove that U ∞ ∈ W ∞ , ∞ . Let 3 < p < ∞ be fixedin the following. We note that(3.5) ( − ∆ + 1) U ∞ = − ∆ (cid:16) C − e − ( V e + ε U ∞ ) (cid:17) + 2 C − e − ( V e + ε U ∞ ) + U ∞ and we study each term in order to prove that this expression is uniformly in L p . Since U ∞ ∈ L ∞ ,we have e − ε U ∞ ∈ L ∞ and we get for all 1 ≤ i, j ≤ ∂ ij ( e − ε U ∞ ) = (cid:0) − ε ∂ ij U ∞ + ε ( ∂ i U ∞ )( ∂ j U ∞ ) (cid:1) e − ε U ∞ ∈ L p N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK uniformly, since on the one hand U ∞ ∈ W ,p uniformly and on the other hand(3.6) ∀ k, ∂ k U ∞ ∈ L p = ⇒ ( ∂ i U ∞ )( ∂ j U ∞ ) ∈ L p . In a direct way we also get ∂ k e − ε U ∞ ∈ L p . Since e − V e ∈ W ,p and using the same trick asin (3.6), this gives from (3.5) that U ∞ ∈ W ,p for the arbitrary fixed 3 < p < ∞ . By a bootstrapargument using the same method we get that U ∞ ∈ W k,p , for all k ∈ N and therefore U ∞ ∈ \ k ∈ N W k,p ⊂ W ∞ , ∞ . The uniformity w.r.t. ε is also clear and the proof of Proposition 3.5 is complete. d = 2. — We consider here only the Coulombian case ( ε > Proposition 3.6 . —
Let d = 2 . Suppose that V e satisfies Assumption 1. Then the uniquesolution U ∞ of the Poisson-Emden equation (3.2) is in W ∞ , ∞ ( R ) , with semi-norms uniformlybounded w.r.t. ε > varying in a small fixed neighbourhood of .Proof . — Notice that when d = 2, the equation (3.2) is equivalent to U ∞ = − π ln | x | ⋆ e − ( V e + ε U ∞ ) . The existence and uniqueness of a solution U ∞ ∈ L p ( R ) for any 1 ≤ p < ∞ with ∇ U ∞ ∈ L ( R )is proved in [ , page 199]. Moreover, the maximum principle ensures that U ∞ ≥
0. It is thenstraightforward to adapt the proof of the case d = 3 to conclude.In the Newtonian case ( ε < V e (e.g. V e ( x ) = | x | , see [ ]),there exist solutions to the equation (3.2), but uniqueness is unknown, even under additionalassumptions on the solution (radial symmetry, regularity, decay at infinity). However it wouldbe interesting to prove the trend to equilibrium also in this case. We refer to [ ], where theauthors obtained such a result for a related problem. Remark 3.7 . — To end this section we notice that since U ∞ ∈ W ∞ , ∞ ( R d ), we get that thepotential at infinity V e + ε U ∞ satisfies the same hypothesis as V e alone. As a consequenceit will be possible to apply to K ∞ all the properties obtained for any generic Fokker-Planckoperator K associated to a generic potential V satisfying Assumptions 1 and 2. This will becrucial in the next section, in which we study the exponential convergence to the equilibrium.A second remark is that the total potential at equilibrium is not explicit. In particular, theGreen function for the equation ∂ t f + K ∞ f is not known. This justifies a posteriori the abstractstudy (anyway with explicit constants) performed in the linear section. In the next section wefirst go on with the study of a generic linear Fokker-Planck operator by studying the long timebehaviour and the exponential decay in time. The aim of thisshort subsection is to prove that we have indeed a uniform estimate on the spectral gap for K ∞ with respect to ε . Let d = 2 or d = 3. We work with the operator K e = v.∂ x − ∂ x V e ( x ) .∂ v − ∂ v . ( ∂ v + v ) FR´ED´ERIC H´ERAU & LAURENT THOMANN and consider a bound from below κ of the spectral gap of W coming from Assumption 2.From [ , Theorem 0.1] we know that there exist constants C , C > t ≥ (cid:13)(cid:13) e − tK e (cid:13)(cid:13) B ⊥ e ≤ C e − tκ /C where B e = (cid:8) f ∈ S ′ ( R d ) s.t. f M − / e ∈ L ( R d ) (cid:9) , and M e is the Maxwellian associated to V e and B ⊥ e is the orthogonal of M e . We then add tothe potential a small perturbation of type εU ∞ with U ∞ ∈ W ∞ , ∞ . This will be applied to thepotential U ∞ built in the preceding subsection. Notice that U ∞ ∈ W ∞ , ∞ with uniform boundswith respect to 0 < ε ≪ K ∞ = v.∂ x − ∂ x V ∞ ( x ) .∂ v − ∂ v . ( ∂ v + v ) , with V ∞ = V e + ε U ∞ . The main result is then the following Proposition 3.8 . —
There exists a small real neighbourhood V of such that for all t ≥ (cid:13)(cid:13) e − tK ∞ (cid:13)(cid:13) B ⊥ ≤ C e − tκ / (8 C ) uniformly w.r.t. ε ∈ V .Proof . — We first recall that in (3.7) the precise result of [ , Theorem 0.1] says that C dependson a finite number of semi-norms of V e and that C = min (cid:8) , κ (cid:9) C e )where C e = max (cid:8) sup (cid:8) Hess( V e ) − ( ( ∂ x V e ) − ∆ V e )Id (cid:9) , (cid:9) . Adding a small perturbation ε U ∞ with U ∞ ∈ W ∞ , ∞ does only change the constant C into 2 C and C into 2 C and we onlyhave to check that κ is changed into κ / ε sufficiently small.For this we look at the spectrum of W ∞ = − ∆ x + | ∂ x V ∞ | / − ∆ x V ∞ / L ( R d ) we have W ∞ = − ∆ x + | ∂ x V ∞ | / − ∆ x V ∞ / ≥ − ∆ x + | ∂ x V e | / − ∆ x V e / ε | ∂ x U ∞ | / − | ε || ∆ U ∞ | / ε ∂ x V e ∂ x U ∞ / ≥ W − κ / ε ∂ x V e ∂ x U ∞ / ε sufficiently small so that ε | ∂ x U ∞ | / | ε || ∆ U ∞ | / ≤ κ /
8. Now there existconstants a and b such that | ∂ x V e ∂ x U ∞ | ≤ aW + b since V e has its second order derivatives bounded, and therefore we get for ε sufficiently small W ∞ ≥ W − κ / . Since W ≥ κ , the minmax principle then directly gives that W ∞ ≥ κ / N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK Remark 3.9 . — We can also notice that the natural norm into the weighted spaces B = (cid:8) f ∈ S ′ s.t. f M − / ∞ ∈ L (cid:9) and B e = (cid:8) f ∈ S ′ s.t. f M − / e ∈ L (cid:9) where M is the Maxwellian associated to V e , are equivalent with an equivalence constantbounded by 1 / ε small enough. This justifies the use of the norms associated tothe space B instead of the one associated to B e in the statement of the main theorems of thisarticle. In the following lemma we crucially use the factthat we work in weighted Sobolev spaces instead of flat ones and that M ∞ ∈ S ( R d ) uniformlyin | ε | ≪
1, as proven in the preceding subsection. We have
Lemma 3.10 . —
Let α ∈ [0 , then there exists C > such that for all h ∈ B α (cid:13)(cid:13)(cid:13) Z h dv (cid:13)(cid:13)(cid:13) H αx ≤ C k h k B α . Proof . — We work by interpolation. Let us first consider the case α = 0. By Cauchy-Schwarz, (cid:13)(cid:13)(cid:13) Z h dv (cid:13)(cid:13)(cid:13) L x = (cid:13)(cid:13)(cid:13) Z h M − / ∞ M / ∞ dv (cid:13)(cid:13)(cid:13) L x ≤ (cid:13)(cid:13)(cid:13)(cid:16) Z h M − ∞ dv (cid:17) / (cid:0) Z M ∞ dv (cid:1) / (cid:13)(cid:13)(cid:13) L x ≤ C k h k B . Now we consider the case α = 1. We write (cid:13)(cid:13)(cid:13) ∂ x Z h dv (cid:13)(cid:13)(cid:13) L x = (cid:13)(cid:13)(cid:13) Z ∂ x h dv (cid:13)(cid:13)(cid:13) L x ≤ (cid:13)(cid:13)(cid:13) Z ( ∂ x + ∂ x V ) h dv (cid:13)(cid:13)(cid:13) L x + (cid:13)(cid:13)(cid:13) Z ( ∂ x V ) h dv (cid:13)(cid:13)(cid:13) L x = (cid:13)(cid:13)(cid:13) Z (cid:0) ( ∂ x + ∂ x V ) h (cid:1) M − / ∞ M / ∞ dv (cid:13)(cid:13)(cid:13) L x + (cid:13)(cid:13)(cid:13) Z h M − / ∞ (cid:0) ∂ x V M / ∞ (cid:1) dv (cid:13)(cid:13)(cid:13) L x ≤ (cid:13)(cid:13)(cid:13)(cid:16) Z (cid:0) ( ∂ x + ∂ x V ) h (cid:1) M − ∞ dv (cid:17) / (cid:0) Z M ∞ dv (cid:1) / (cid:13)(cid:13)(cid:13) L x + (cid:13)(cid:13)(cid:13)(cid:16) Z h M − ∞ dv (cid:17) / (cid:16) Z ( ∂ x V ) M ∞ dv (cid:17) / (cid:13)(cid:13)(cid:13) L x ≤ C (cid:13)(cid:13)(cid:13) ( ∂ x + ∂ x V ) h (cid:13)(cid:13)(cid:13) B + C k h k B ≤ C k h k B where we used that ( ∂ x V ) M ∞ ∈ L ∞ x L v , and that k ( ∂ x + ∂ x V ) h k B + k h k B = ( − ∂ x ( ∂ x + ∂ x V ) h , h ) B + k h k B = (Λ x h , h ) B = k Λ x h k B ≤ k h k B . This gives the result for α = 1. The complete result follows by interpolation. Lemma 3.11 . —
Assume that d = 2 or d = 3 . Let h ∈ B and denote by E ( x ) = x | x | d ⋆ Z h ( x, v ) dv. (i) Case d = 2 . For all < ε ≤ / there exists C > so that (3.9) k E k L ∞ ( R ) ≤ C k h k B ε . (ii) Case d = 3 . For all < ε ≤ / there exists C > so that (3.10) k E k L ∞ ( R ) ≤ C k h k B / ε . FR´ED´ERIC H´ERAU & LAURENT THOMANN
Proof . — Let us first recall the Hardy-Littlewood-Sobolev inequality (see e.g. [ ]) which willbe useful in the sequel. For all 1 < p, q < + ∞ such that q − p + d = 0(3.11) (cid:13)(cid:13) x | x | d ⋆ f (cid:13)(cid:13) L q ( R d ) ≤ C k f k L p ( R d ) . We prove ( ii ). We consider the Fourier multiplier L x = (1 − ∆ x ) / . Then, by Hardy-Littlewood-Sobolev and the Sobolev embeddings, for any ε > k E k L ∞ ( R ) ≤ C (cid:13)(cid:13) L εx Z h dv (cid:13)(cid:13) L ( R ) ≤ C (cid:13)(cid:13) L / εx Z h dv (cid:13)(cid:13) L ( R ) ≤ (cid:13)(cid:13) Z h dv (cid:13)(cid:13) H / ε ( R ) . Using Lemma 3.10 with α = 1 / ε we get (3.10).The proof of ( i ) is analogous with L εx replaced with L / εx . Corollary 3.12 . —
Assume that d = 2 or d = 3 . Let f ∈ B ⊥ and denote by E ( t, x ) = x | x | d ⋆ Z e − tK f ( x, v ) dv. (i) Case d = 2 . For all < ε ≤ / there exists C > so that for all t > k E ( t, . ) k L ∞ ( R ) ≤ C (1 + t − ε/ ) e − κt k f k B . (ii) Case d = 3 . For all < ε ≤ / there exists C > so that for all t > k E ( t, . ) k L ∞ ( R ) ≤ Ce − κt k f k B / ε . Proof . — ( i ). We apply the result of Lemma 3.11 to the case h = e − tK f for some f ∈ B ⊥ ,then(3.14) k E ( t, . ) k L ∞ ( R ) ≤ C k Λ εx e − tK f k B . Thus estimate (2.7) together with (3.14) implies k E ( t, . ) k L ∞ ( R ) ≤ C (1 + t − ε/ ) e − κt k f k B , which was to prove.( ii ). By (3.10) and (2.11), we obtain k E ( t, . ) k L ∞ ( R ) ≤ C k Λ / ε e − tK Λ − / − ε k B ⊥ → B ⊥ k Λ / ε f k B ≤ Ce − κt k f k B / ε , which was the claim. In this subsection we give a technical result.
Lemma 3.13 . —
Let γ , γ , c > and assume that γ ≤ . Then there exists C > so that forall t > Z t (cid:0) s − γ + 1 (cid:1)(cid:0) ( t − s ) − γ + 1 (cid:1) e − c ( t − s ) ds ≤ ( C (cid:0) t − γ + γ + 1 (cid:1) for t ≤ ,C for t ≥ . Proof . — The proof is elementary: we expand the r.h.s. of (3.15) and estimate each piece. Let t ≤
1, then Z t (cid:0) s − γ + 1 (cid:1)(cid:0) ( t − s ) − γ + 1 (cid:1) e − c ( t − s ) ds ≤ Z t (cid:0) s − γ + 1 (cid:1)(cid:0) ( t − s ) − γ + 1 (cid:1) ds, Firstly, Z t s − γ ( t − s ) − γ ds = C γ ,γ t − γ + γ , N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK by a simple change of variables. Then for t ≤ Z t s − γ ds + Z t ( t − s ) − γ ds ≤ C, and this yields the result. Now we assume that t ≥
1. Then on the one hand Z (cid:0) s − γ + 1 (cid:1)(cid:0) ( t − s ) − γ + 1 (cid:1) e − c ( t − s ) ds ≤ C e − ct Z (cid:0) s − γ + 1 (cid:1)(cid:0) ( t − s ) − γ + 1 (cid:1) ds ≤ C, and on the other hand, since γ ≤ Z t (cid:0) s − γ + 1 (cid:1)(cid:0) ( t − s ) − γ + 1 (cid:1) e − c ( t − s ) ds ≤ C Z t (cid:0) ( t − s ) − γ + 1 (cid:1) e − c ( t − s ) ds ≤ C, which completes the proof. In this subsection we show how some of the previousresults on the Fokker-Planck operator with potential satisfying Assumption 1 remain valid whenthe potential is of type V = V e + ε U where V e satisfies Assumption 1, U ∈ W , ∞ and | ε | ≤
1. This will be applied in Section 5when the study for short time will be done.In the following we denote by K = v.∂ x − ∂ x V e ( x ) .∂ v − ∂ v . ( ∂ v + v )and K = K − ε ∂ x U ( x ) .∂ v . Note that the Hilbert spaces of type B defined in (1.5) with either M ∞ (defined in (1.2)) or M e (when V e + ε U ∞ is replaced there by V e only) or even M (when V e + ε U ∞ is replacedthere by V e + ε U ) are all equal with equivalent norms uniformly in 0 ≤ ε ≤ U or U ∞ .We will need the following result Lemma 3.14 . —
The domains of K and K coincide, they are both maximal accretive with M / S as a core.Proof . — This is clear for K as already noticed and used (see [ ]). The difficulty is that K hasonly W , ∞ coefficients. There exists C > k ∂ x U k L ∞ ≤ C , and then for any η > C η > k ∂ x U .∂ v f k B ≤ C k ∂ v f k B ≤ η k Kf k B + C η k f k B , which directly implies that the domains are the same, see e.g. [ , Chapter III, Lemma 2.4].The fact that M / S is a core is also a direct consequence of this inequality.We now prove that some results from Section 2 about semigroup estimates remain true forthe new operator K with non-smooth coefficients.We begin with a general Proposition Proposition 3.15 . —
Let us consider the operator K with potential V e + ε U . Then thereexists C > such that the following is true uniformly in ε ∈ [0 , and t ∈ (0 , FR´ED´ERIC H´ERAU & LAURENT THOMANN (i) ∀ γ ∈ [0 , , (cid:13)(cid:13) Λ γ e − tK Λ − γ (cid:13)(cid:13) B → B ≤ C ,(ii) ∀ β ∈ [0 , , (cid:13)(cid:13)(cid:13) Λ βv e − tK (cid:13)(cid:13)(cid:13) B → B ≤ C t − β/ ,(iii) ∀ α ∈ [0 , , (cid:13)(cid:13) Λ αx e − tK (cid:13)(cid:13) B → B ≤ C t − α/ ,(iv) ∀ a ∈ [0 , and f ∈ B a , (cid:13)(cid:13) ( e − tK − f (cid:13)(cid:13) B ≤ C t a k f k B a . Proof . — We first note that the proof of point ( iv ) given in Lemma 2.9 is unchanged (for a = 0)under the new assumptions on the potential V , and uniformly w.r.t. ε . For points ( iii ) and ( ii )this is the same w.r.t. the proof of Proposition 2.1 and we emphasise that the constants onlydepend on the second derivatives of the potential, which are here uniformly bounded w.r.t. ε .It therefore only remains to check point ( i ) for which the proof of point (2.11) cannot bedirectly adapted, since we have to restrict here to the case when γ ∈ [0 , e − tK is bounded from B γ,γx,v into itself. We first begin with the case γ = 1. We now use that k f k B ∼ k Λ f k B ∼ k ( ∂ x + ∂ x V ) f k B + k ( ∂ v + v ) f k B with uniform w.r.t. ε equivalence constants, since U ∈ W , ∞ . We therefore look, for an initialdata f ∈ B , x,v at the equation satisfied by g = ( ∂ x + ∂ x V ) f and h = ( ∂ v + v ) f in B . We consideragain the operator X = v.∂ x − ∂ x V e .∂ v . Since ∂ t f + X f − ε ∂ x U .∂ v f − ∂ v . ( ∂ v + v ) f = 0 , f t =0 = f we get the system ∂ t g + X g − ε ∂ x U ∂ v g − ∂ v . ( ∂ v + v ) g = Hess V h∂ t h + X h − ε ∂ x U ∂ v h − ∂ v . ( ∂ v + v ) h = − h − g + ε ∂ x U f, with g t =0 = g ∈ B and h t =0 = h ∈ B. Integrating the three last equations against respectively f , g and h in B gives, ∂ t ( k f k B + k g k B + k h k B ) ≤ C ( k f k B + k g k B + k h k B )since V has a Hessian uniformly bounded w.r.t. ε . We therefore get k f ( t ) k B + k g ( t ) k B + k h ( t ) k B ≤ C e C t ( k f k B + k g k B + k h k B )and we get that e − tK is (uniformly in t ∈ [0 ,
1] and ε ∈ [0 , B to B . Nowthe result is also clear for γ = 0 by the semi group property, and by interpolation we get that e − tK is (uniformly in t ∈ [0 ,
1] and ε ∈ [0 , B γ to B γ for γ ∈ [0 , k Λ γ e − tK Λ − γ k B → B ≤ C γ . This concludes the proof of point ( i ) and the proof of the Proposition.As a consequence, a certain number of results of Section 2 remain true with proofs withoutchanges. We gather them in the following corollary. Corollary 3.16 . —
There exists
C > such that the following is true uniformly in ε ∈ [0 , and t ∈ (0 , (i) ∀ β ∈ [0 , , ∀ a ∈ [0 , β ] , (cid:13)(cid:13)(cid:13) Λ βv e − tK (cid:13)(cid:13)(cid:13) B a → B ≤ C (1 + t − ( β − a ) / ) ,(ii) ∀ β ∈ [0 , , (cid:13)(cid:13)(cid:13) Λ βv e − tK Λ − βv (cid:13)(cid:13)(cid:13) B → B ≤ C (1 + t − / ) ,(iii) ∀ α, β ∈ [0 , , (cid:13)(cid:13)(cid:13) Λ α e − tK Λ − βv (cid:13)(cid:13)(cid:13) B → B ≤ C (1 + t − / β/ − α/ ) . N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK Proof . — The proof of ( i ) follows the one of Lemma 2.8 thanks to points ( i ), ( ii ) and ( iii )in Proposition 3.15. Points ( ii ) and ( iii ) are consequences respectively of ( ii ) and ( iii ) ofProposition 3.15 since (cid:13)(cid:13)(cid:13) Λ βv e − tK Λ − βv (cid:13)(cid:13)(cid:13) B → B ≤ (cid:13)(cid:13)(cid:13) Λ βv e − tK / (cid:13)(cid:13)(cid:13) B → B (cid:13)(cid:13)(cid:13) e − tK / Λ − βv (cid:13)(cid:13)(cid:13) B → B and (cid:13)(cid:13)(cid:13) Λ α e − tK Λ − βv (cid:13)(cid:13)(cid:13) B → B ≤ (cid:13)(cid:13)(cid:13) Λ α e − tK / (cid:13)(cid:13)(cid:13) B → B (cid:13)(cid:13)(cid:13) e − tK / Λ − βv (cid:13)(cid:13)(cid:13) B → B . Remark 3.17 . — Let us observe that if one only has f ∈ B ( R ) ∩ L ∞ ( R ), one can provethat U defined in (1.10) satisfies U ∈ W ,p ( R ) for any 2 ≤ p < ∞ . In other words, theassumption U ∈ W , ∞ ( R ) fills in an ε − gap of regularity. More precisely, let p ≥ q ≤ /p + 1 /q = 1. Then, by H¨older | ∆ U | = Z f dv ≤ (cid:0) Z f p M − dv (cid:1) /p (cid:0) Z M q/p dv (cid:1) /q . Thus using that Z M q/p dv ∈ L ∞ ( R ), we get Z | ∆ U | p dx ≤ C Z f p M − dvdx ≤ C k f k p − L ∞ ( R ) k f k B , which implies that U ∈ W ,p ( R ) by elliptic regularity.Now we prove a result that will be useful for the short time analysis in the next section. Againwe work with the linear Fokker-Planck operator K with potential V e + ε U . Lemma 3.18 . —
Assume that d = 3 and a > / . Let f ∈ B a ( R ) ∩ L ∞ ( R ) and denote by S ( t, x ) = x | x | ⋆ Z (cid:0) e − tK − (cid:1) f ( x, v ) dv. Then for all ε ≪ and ≤ t ≤ and uniformly in ε ∈ [0 , we have (3.16) k S ( t ) k L ∞ ( R ) ≤ Ct a/ − ε (cid:0) k f k L ∞ + k f k B a (cid:1) . Proof . — In the sequel, 0 ≤ t ≤ σ = a − / > q > /σ be large. Thenby the Gagliardo-Nirenberg inequality(3.17) k S k L ∞ x ≤ C k S k − σq L qx k S k σq W σ,qx , and we now estimate the previous terms.By (3.11), there exists p < p −→ q −→ + ∞ ) such that k S k L qx ≤ C (cid:13)(cid:13) Z h dv (cid:13)(cid:13) L px , where h = (cid:0) e − tK − (cid:1) f . Then, by H¨older (where p ′ is the conjugate of p ) Z | h | dv = Z (cid:0) | h |M − /p ∞ (cid:1) M /p ∞ dv ≤ (cid:16) Z | h | p M − ∞ dv (cid:17) /p (cid:16) Z M p ′ /p ∞ dv (cid:17) /p ′ ≤ C (cid:16) Z | h | p M − ∞ dv (cid:17) /p . This implies that(3.18) k S k L qx ≤ C (cid:13)(cid:13) Z h dv (cid:13)(cid:13) L px ≤ C (cid:16) Z | h | p M − ∞ dvdx (cid:17) /p ≤ C k h k − /pL ∞ k h k /pB . FR´ED´ERIC H´ERAU & LAURENT THOMANN
Now, by point ( iv ) of Proposition 3.15 we have k h k L ∞ ≤ C k f k L ∞ , and by Lemma 2.9, k h k B ≤ Ct a/ k f k B a , hence k S k L qx ≤ Ct a/p (cid:0) k f k L ∞ + k f k B a (cid:1) . Next, by (3.11) and Sobolev (recall that p ∼ q large) k S k W σ,qx ≤ C (cid:13)(cid:13) (1 − ∆ x ) σ/ Z h dv (cid:13)(cid:13) L px ≤ C (cid:13)(cid:13) (1 − ∆ x ) a/ Z h dv (cid:13)(cid:13) L x , since σ + 1 / a . Then we proceed as in the proof of (3.10) to get(3.19) k S k W σ,qx ≤ C k h k B a ≤ C k f k B a . Fix ε ≪
1. Then for q ≫
1, we combine (3.17), (3.18) and (3.19) to get (3.16).
4. Proof of Theorem 1.2 (case d = 2)4.1. Functional setting. — To begin with, we introduce the functional framework which willbe used in both cases d = 2 or d = 3.To show the trend to equilibrium, we look for a solution of the form f = f ∞ + g with f ∞ = c M ∞ and g ∈ B ⊥ . The normalization R f dxdv = R M ∞ dxdv = 1 then implies that f ∞ = M ∞ . Hence we write f = M ∞ + g, E = E ∞ + F, with ∂ x U ∞ = E ∞ = − | S d − | x | x | d ⋆ Z M ∞ dv, F = − | S d − | x | x | d ⋆ Z gdv. In the sequel denote by K = K ∞ . We want to take profit of the regularization property stated in Lemma 3.11, thus we look for asolution of the form g = e − tK g + h, F = F + G, with F = − | S d − | x | x | d ⋆ Z e − tK g dv, G = − | S d − | x | x | d ⋆ Z hdv, and h (0) = G (0) = 0. At this stage we observe that f = M ∞ + g and that for all t ≥ − tK f = M ∞ + e − tK g .We construct the solution with a fixed point argument on ( h, G ), and therefore we define themap Φ = (Φ , Φ ) given byΦ ( h, G )( t ) = ε Z t e − ( t − s ) K (cid:0) F ( s ) + G ( s ) (cid:1) ∂ v (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) ds Φ ( h, G )( t ) = − ε | S d − | x | x | d ⋆ Z R d Z t e − ( t − s ) K (cid:0) F ( s ) + G ( s ) (cid:1) ∂ v (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) dsdv, and we observe that ( f, E ) solves (1.9) if and only if ( h, G ) = Φ( h, G ). For α, β, γ, δ, σ ≥ k h k X α,βδ = sup t ≥ (cid:16) t δ t δ e σκt k h ( t, . ) k B α,βx,v ( R d ) (cid:17) , k G k Y = sup t ≥ (cid:16) e σκt k G ( t, . ) k L ∞ ( R d ) (cid:17) , N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK define the Banach space Z := X α,βδ × Y, with k ( h, G ) k Z = max (cid:0) k h k X α,βδ , k G k Y (cid:1) , and denote by Γ its unit ball. In each of the cases d = 2 or 3, for a given initial condition g ,we will prove that if | ε | < ⊂ Z . Toalleviate notations, we assume in the sequel that ε > d = 2. — This case is the easiest. Let g ∈ B .We can fix here α = β = δ = 0. Let ε ≪ σ = 1 /
2. For simplicity, we write X = X , .We proceed in two steps. Recall that Γ is the unit ball of Z . Then Step1: Φ maps the ball Γ ⊂ Z into itself • We estimate Φ ( h, G ) in X . By (2.13) and (2.6), we have for all t ≥ k Φ ( h, G )( t ) k B ≤ ε Z t k e − ( t − s ) K (cid:0) F + G (cid:1) ∂ v (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) k B ds ≤ Cε Z t k F + G k L ∞ ( R ) k e − ( t − s ) K Λ v k B ⊥ kM ∞ + e − sK g + h ( s ) k B ds, (4.1)and we estimate each factor in the previous integral.Estimation of kM ∞ + e − sK g + h ( s ) k B : We use that M ∞ ∈ B , and by (2.1) we obtain kM ∞ + e − sK g + h ( s ) k B ≤ kM ∞ k B + k e − sK g k B + k h ( s ) k B ≤ C (1 + k h k X ) . (4.2)Estimation of k F + G k L ∞ ( R ) : By (3.12) we get k F + G k L ∞ ( R ) ≤ k F k L ∞ ( R ) + k G k L ∞ ( R ) ≤ C (1 + s − ε/ ) e − σκs k g k B + C e − σκs k G k Y ≤ C (1 + s − ε/ )e − σκs (1 + k G k Y ) . (4.3)Estimation of k e − ( t − s ) K Λ v k B ⊥ → B ⊥ : This follows from (2.8)(4.4) k e − ( t − s ) K Λ v k B ⊥ → B ⊥ ≤ C (cid:0) ( t − s ) − / + 1 (cid:1) e − κ ( t − s ) . Therefore by (4.1), (4.2), (4.3) and (4.4) we have k Φ ( h, G )( t ) k B ≤ Cε (1 + k h k X )(1 + k G k Y )e − σκt Z t (cid:0) s − ε/ + 1 (cid:1)(cid:0) ( t − s ) − / + 1 (cid:1) e − κ ( t − s ) / ds. Now, by (3.15) we deduce k Φ ( h, G )( t ) k B ≤ Cε (1 + k h k X )(1 + k G k Y )e − σκt , which in turn yields the bound(4.5) k Φ ( h, G ) k X ≤ Cε (1 + k h k X )(1 + k G k Y ) ≤ Cε (cid:0) k ( h, G ) k Z (cid:1) . • We turn to the estimation of k Φ ( h, G ) k Y . We apply (3.9) with h = Z t e − ( t − s ) K (cid:0) F ( s ) + G ( s ) (cid:1) ∂ v (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) ds, FR´ED´ERIC H´ERAU & LAURENT THOMANN then for all t ≥ k Φ ( h, G )( t ) k L ∞ ( R ) ≤ C k Λ ε Z t e − ( t − s ) K (cid:0) F ( s ) + G ( s ) (cid:1) ∂ v (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) ds k B ≤ C Z t k Λ ε e − ( t − s ) K (cid:0) F + G (cid:1) ∂ v (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) k B ds ≤ Cε Z t k F + G k L ∞ ( R ) k Λ ε e − ( t − s ) K Λ v k B ⊥ kM ∞ + e − sK g + h ( s ) k B ds, where in the last line we used (2.13). Then by (4.2), (4.3) and (2.9) with α = ε and β = 0 weget k Φ ( h, G )( t ) k L ∞ ( R ) ≤≤ Cε (1 + k h k X )(1 + k G k Y )e − σκt Z t (cid:0) s − ε/ + 1 (cid:1)(cid:0) ( t − s ) − / − ε/ + 1 (cid:1) e − κ ( t − s ) / ds. By (3.15), this in turn implies(4.6) k Φ ( h, G ) k Y ≤ Cε (1 + k h k X )(1 + k G k Y ) ≤ Cε (cid:0) k ( h, G ) k Z (cid:1) . As a result, by (4.5) and (4.6) there exists
C > k Φ( h, G ) k Z ≤ Cε (cid:0) k ( h, G ) k Z (cid:1) . Therefore we can choose ε > ⊂ Z into itself. Step2: Φ is a contraction of Γ With exactly the same arguments, we can also prove the contraction estimate k Φ ( h , G ) − Φ ( h , G ) k Z ≤ Cε k ( h − h , G − G ) k Z (cid:0) k ( h , G ) k Z + k ( h , G ) k Z (cid:1) . We do not write the details.As a conclusion, if ε > ⊂ Z . This showsthe existence of a unique h ∈ C (cid:0) [0 , + ∞ [ ; B ( R ) (cid:1) such that f = M ∞ + e − tK g + h solves (1.1). The convergence of f to equilibriumfollows from the choice of the weighted spaces. By definition k h ( t ) k B ≤ C e − σκt k h k X −→ , when t −→ + ∞ . Similarly, k G ( t ) k L ∞ ≤ C e − σκt k G k Y −→ , when t −→ + ∞ .
5. Proof of Theorem 1.3 (case d = 3)5.1. Small time analysis: ≤ t ≤ . — To begin with we prove a local well-posedness resultfor (1.1).
Proposition 5.1 . —
Let d = 3 and / < a < / . Assume that f ∈ B a,a ( R ) ∩ L ∞ ( R ) issuch that U defined in (1.10) is in W , ∞ ( R ) . Assume moreover that Assumptions 1 and 2 aresatisfied. Then if | ε | is small enough, there exists a unique local mild solution f to (1.1) in theclass f ∈ C (cid:0) [0 ,
1] ; B a,a ( R ) (cid:1) ∩ L ∞ (cid:0) [0 ,
1] ; L ∞ ( R ) (cid:1) . N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK We write f = e − tK f + g, E = E + F, where U , E and F are defined by ∂ x U = E = − | S | x | x | ⋆ Z f dv, F = − | S | x | x | ⋆ Z gdv. In the regime 0 ≤ t ≤
1, the effective Fokker-Planck operator is given by K = v.∂ x − ∂ x V ( x ) .∂ v − ∂ v . ( ∂ v + v ) , where V = V e + ε U . The mild formulation of (1.1), using K , is therefore(5.1) f ( t ) = e − tK f + ε Z t e − ( t − s ) K ( E ( s ) − E ) ∂ v f ( s ) ds,E ( t ) = − | S | x | x | ⋆ x Z f ( t ) dv. We construct the solution with a fixed point argument on ( g, F ), and therefore we define themap Φ = (Φ , Φ ) given byΦ ( g, F )( t ) = ε Z t e − ( t − s ) K F ( s ) ∂ v (cid:0) e − sK f + g ( s ) (cid:1) ds Φ ( g, F )( t ) = − | S | x | x | ⋆ Z R h(cid:0) e − tK − (cid:1) f + ε Z t e − ( t − s ) K F ( s ) ∂ v (cid:0) e − sK f + g ( s ) (cid:1) ds i , and we observe that ( f, E ) solves (5.1) if and only if ( g, F ) = Φ( g, F ). For α, β, γ ≥ k g k X α,β = sup ≤ t ≤ k g ( t, . ) k B α,βx,v ( R ) , k F k Y γ = sup ≤ t ≤ t − γ k F ( t, . ) k L ∞ ( R ) , define the Banach space Z := X α,β × Y γ , with k ( h, G ) k Z = max (cid:0) k h k X α,β , k G k Y γ (cid:1) , and denote by Γ R the ball of radius R .In the sequel we fix γ = a/ − ε, α = a, β = 1 , for some ε ≪ g ∈ B a,a , for some a > /
2. In the sequel, we write K = K . Step1: Φ maps some ball Γ R ⊂ Z into itself • Firstly, we estimate Φ ( g, F ) in X , . By (2.13), we have for all 0 ≤ t ≤ k Λ v Φ ( g, F )( t ) k B ≤ ε Z t k Λ v e − ( t − s ) K F ( s ) ∂ v (cid:0) e − sK f + g ( s ) (cid:1) k B ds ≤ Cε Z t k F k L ∞ ( R ) k Λ v e − ( t − s ) K k B k Λ v (cid:0) e − sK f + g ( s ) (cid:1) k B ds, (5.2)and we estimate each factor in the previous integral thanks to the low regularity subsectionresults.Estimation of k Λ v (cid:0) e − sK f + g ( s ) (cid:1) k B : To begin with, we use point ( i ) of Corollary 3.16 toestimate k Λ v e − sK f k B . Since f ∈ B a,a for some a > /
2, then for δ = 1 / − a/ k Λ v e − sK f k B ≤ Cs − δ k f k B a . FR´ED´ERIC H´ERAU & LAURENT THOMANN
This gives for 0 ≤ s ≤ t ≤ k Λ v (cid:0) e − sK f + g ( s ) (cid:1) k B ≤ k e − sK f k B , + k g ( s ) k B , ≤ C + Cs − δ k f k B a + k g k X a, ≤ Cs − δ (1 + k g k X a, ) . (5.3)Estimation of k F k L ∞ ( R ) : By definition of the space Y γ we have(5.4) k F k L ∞ ( R ) ≤ s γ k F k Y γ . Estimation of k Λ v e − ( t − s ) K k B → B : By point ( ii ) in Proposition 3.15 we have(5.5) k Λ v e − ( t − s ) K k B → B ≤ C ( t − s ) − / . Therefore by (5.2), (5.3), (5.4) and (5.5), we have k Λ v Φ ( g, F )( t ) k B ≤ Cε (1 + k g k X a, ) k F k Y γ Z t s γ − δ ( t − s ) − / ds ≤ Cε (1 + k g k X a, ) k F k Y γ t γ − δ +1 / . As a consequence (using that γ − δ + 1 / ≥
0) we have proved(5.6) k Φ ( g, F ) k X , ≤ Cε (1 + k g k X a, ) k F k Y γ ≤ Cε (cid:0) k ( g, F ) k Z (cid:1) . • We estimate Φ ( g, F ) in X a, . With the same arguments and the bound given in point ( iii )of Corollary 3.16, for all t ≥ k Λ ax Φ ( g, F )( t ) k B ≤ ε Z t k Λ ax e − ( t − s ) K F ( s ) ∂ v (cid:0) e − sK f + g ( s ) (cid:1) k B ds (5.7) ≤ Cε Z t k F k L ∞ ( R ) k Λ ax e − ( t − s ) K k B k Λ v (cid:0) e − sK f + g ( s ) (cid:1) k B ds ≤ Cε (1 + k g k X a, ) k F k Y γ Z t s γ − δ (cid:0) ( t − s ) − a/ + 1 (cid:1) ds ≤ Cε (1 + k g k X a, ) k F k Y γ (cid:0) t γ − δ +1 − a/ + t γ − δ +1 (cid:1) . This in turn implies (observing that γ − δ + 1 − a/ > a < / k Φ ( g, F ) k X a, ≤ Cε (1 + k g k X a, ) k F k Y γ ≤ Cε (cid:0) k ( g, F ) k Z (cid:1) . • We turn to the estimation of k Φ ( g, F ) k Y γ . We apply (3.16) and (3.10) with h = (cid:0) e − tK − (cid:1) f + ε Z t e − ( t − s ) K F ( s ) ∂ v (cid:0) e − sK f + g ( s ) (cid:1) ds, then for all 0 ≤ t ≤ k Φ ( g, F )( t ) k L ∞ ( R ) ≤ C t a/ − ε + Cε Z t k Λ / ε e − ( t − s ) K F ( s ) ∂ v (cid:0) e − sK f + g ( s ) (cid:1) k B ds, where we used Lemma 3.18.To control the second term, we can proceed as in (5.7) with a replaced by 1 / ε . Actuallywe have k Φ ( g, F )( t ) k L ∞ ( R ) ≤ C t a/ − ε + Cε Z t k F k L ∞ ( R ) k Λ / ε e − ( t − s ) K k B k Λ v (cid:0) e − sK f + g ( s ) (cid:1) k B ds, N GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK and we get k Φ ( g, F )( t ) k L ∞ ( R ) ≤ C t a/ − ε + Cε (1 + k g k X a, ) k F k Y γ Z t s γ − δ (cid:0) ( t − s ) − ( + ε ) + 1 (cid:1) ds ≤ C t a/ − ε + Cε (1 + k g k X a, ) k F k Y γ (cid:0) t γ − δ +1 / − ε/ + t γ − δ +1 (cid:1) ≤ t γ (cid:2) C + Cε (1 + k g k X a, ) k F k Y γ ( t − δ +1 / − ε/ + t − δ +1 ) (cid:3) , since γ = a/ − ε . Therefore(5.9) k Φ ( g, F ) k Y γ ≤ C + Cε (1 + k g k X a, ) k F k Y γ ≤ Cε (cid:0) k ( g, F ) k Z (cid:1) , provided that δ < / − a/ δ < / − ε/
2. This latter condition can be satisfied for ε > a > / C > k Φ( g, F ) k Z ≤ C + Cε (cid:0) k ( g, F ) k Z (cid:1) . Therefore we can choose ε > C ⊂ Z into itself. Step2: Φ is a contraction of Γ C With exactly the same arguments, we can also prove the contraction estimate k Φ ( g , F ) − Φ ( g , F ) k Z ≤ Cε k ( g − g , F − F ) k Z (cid:0) k ( g , F ) k Z + k ( g , F ) k Z (cid:1) . We do not write the details.As a conclusion, if ε > C ⊂ Z . Thisshows the existence of a unique g ∈ C (cid:0) [0 ,
1] ; B a, ( R ) (cid:1) such that f = e − tK f + g solves (1.1). t ∈ ]0 , + ∞ [ . — We now study long time existence and trend toequilibrium. We use here the spaces defined in the Subsection 4.1. Let 1 / < β < < α < / α < (1 + β ) /
3. Fix also0 < δ < β/ − / , < σ ≤
12 min (cid:0) − β + α, (cid:1) , which is realised for, say, σ = 1 /
12. From now, we assume that all these conditions are satisfied.In this section we prove the following result
Proposition 5.2 . —
Let d = 3 . Assume that / < a < / and that f ∈ B a,a ( R ) . Assumemoreover that Assumptions 1 and 2 are satisfied. Then if | ε | is small enough, there exists aunique local mild solution f to (1.1) which reads f ( t ) = M ∞ + e − tK ( f − M ∞ ) + h ( t ) where h ∈ X α,βδ , thus h ∈ C (cid:0) ]0 , + ∞ [ ; B α,β ( R ) (cid:1) . Since f − M ∞ ∈ B ⊥ , and by definition of the space X α,βδ , we obtain the exponentially fastconvergence of f to M ∞ . Notice that in the previous result, the parameters ( α, β ) can be chosenindependently from a . If one chooses α = a and β close to 1, then the result of Proposition 5.2combined with Proposition 5.1 and Corollary 3.2 implies Theorem 1.3.We now turn to the proof of Proposition 5.2. Let g := f − M ∞ ∈ B a,a ∩ B ⊥ , for some a > /
2. We denote by Γ the unit ball in Z , and in the sequel, we use the same notations anddecomposition as in Section 4.1. Then FR´ED´ERIC H´ERAU & LAURENT THOMANN
Step1: Φ maps the ball Γ ⊂ Z into itself • Firstly, we estimate Φ ( h, G ) in X ,βδ . By (2.13), we have for all t ≥ k Λ βv Φ ( h, G )( t ) k B ≤ ε Z t k Λ βv e − ( t − s ) K (cid:0) F + G (cid:1) ∂ v (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) k B ds ≤ Cε Z t k F + G k L ∞ ( R ) k Λ βv e − ( t − s ) K Λ − βv k B ⊥ k Λ βv (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) k B ds, and we estimate each factor in the previous integral.Estimation of k Λ βv (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) k B : To begin with, we use Lemma 2.8 to estimatethe term k Λ βv e − sK g k B . Since g ∈ B a,a for some a > /
2, there exists 0 < δ < β/ − / k Λ βv e − sK g k B ≤ C (1 + s − δ ) k g k B a . Then, we use that M ∞ ∈ B α,β , and by (2.8) we obtain k Λ βv (cid:0) M ∞ + e − sK g + h ( s ) (cid:1) k B ≤ kM ∞ k B ,β + k e − sK g k B ,β + k h ( s ) k B ,β ≤ C + C (1 + s − δ ) k g k B a + (1 + s − δ ) k h k X α,βδ ≤ C (1 + s − δ )(1 + k h k X α,βδ ) . (5.11)Estimation of k F + G k L ∞ ( R ) : By (3.13) and the definition of the space Y we get for ε > a > / k F + G k L ∞ ( R ) ≤ k F k L ∞ ( R ) + k G k L ∞ ( R ) ≤ Ce − κs/ k g k B a + C e − σκs k G k Y ≤ C e − σκs (1 + k G k Y ) . (5.12)Estimation of k Λ βv e − ( t − s ) K Λ − βv k B ⊥ → B ⊥ : This is exactly (2.10), namely k Λ βv e − ( t − s ) K Λ − βv k B ⊥ → B ⊥ ≤ C (cid:0) ( t − s ) − / + 1 (cid:1) e − κ ( t − s ) . Therefore by (5.10), (5.11), (5.12), we have when δ < k Λ βv Φ ( h, G )( t ) k B ≤≤ Cε (1 + k h k X α,βδ )(1 + k G k Y )e − σκt Z t (cid:0) s − δ + 1 (cid:1)(cid:0) ( t − s ) − / + 1 (cid:1) e − κ ( t − s ) / ds Now, by (3.15), if we denote by η = − min(1 / − δ, k Λ βv Φ ( h, G )( t ) k B ≤ Cε (1 + k h k X α,βδ )(1 + k G k Y )e − σκt (cid:0) {
1, by (2.9) and (3.15) we get k Φ ( h, G )( t ) k L ∞ ( R ) ≤≤ Cε (1 + k h k X α,βδ )(1 + k G k Y )e − σκt Z t (cid:0) s − δ + 1 (cid:1)(cid:0) ( t − s ) − + β − ( + ε ) + 1 (cid:1) e − κ ( t − s ) / ds ≤ Cε (1 + k h k X α,βδ )(1 + k G k Y )e − σκt (cid:0) {
2. This in turn implies(5.17) k Φ ( h, G ) k Y ≤ Cε (1 + k h k X α,βδ )(1 + k G k Y ) ≤ Cε (cid:0) k ( h, G ) k Z (cid:1) . As a result, by (5.14), (5.16) and (5.17) there exists
C > k Φ( h, G ) k Z ≤ Cε (cid:0) k ( h, G ) k Z (cid:1) . Therefore we can choose ε > ⊂ Z into itself. Step2: Φ is a contraction of Γ FR´ED´ERIC H´ERAU & LAURENT THOMANN
With exactly the same arguments, we can also prove the contraction estimate k Φ ( h , G ) − Φ ( h , G ) k Z ≤ Cε k ( h − h , G − G ) k Z (cid:0) k ( h , G ) k Z + k ( h , G ) k Z (cid:1) . We do not write the details.As a conclusion, if ε > ⊂ Z . This showsthe existence of a unique h ∈ C (cid:0) ]0 , + ∞ [ ; B α,β ( R ) (cid:1) such that f = M ∞ +e − tK g + h solves (1.1). References [1] A. Blanchet, J. Dolbeault and B. Perthame. Two-dimensional Keller-Segel model: optimal criticalmass and qualitative properties of the solutions.
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Fr´ed´eric H´erau , Laboratoire de Math´ematiques J. Leray, UMR 6629 du CNRS, Universit´e de Nantes, 2, ruede la Houssini`ere, 44322 Nantes Cedex 03, France • E-mail : [email protected]