FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES WITH POLYNOMIAL WEIGHT

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1 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES WITH POLYNOMIAL WEIGHT M BRIANT, S MERINO-ACEITUNO, C MOUHOT Abstract We stuy the Boltzmann euation on the -imensional torus in a perturbatie setting aroun a global euilibrium uner the Naier-Stoes linearisation We use a recent functional analysis breathrough to proe that the linear part of the euation generates a C -semigroup with exponential ecay in Lebesgue an Sobole spaces with polynomial weight, inepenently on the Knusen number Finally we show a Cauchy theory an an exponential ecay for the perturbe Boltzmann euation, uniformly in the Knusen number, in Sobole spaces with polynomial weight The polynomial weight is almost optimal an furthermore, this result only reuires eriaties in the space ariable an allows to connect to solutions to the incompressible Naier-Stoes euations in these spaces Keywors: Boltzmann euation on the Torus; Incompressible Naier-Stoes hyroynamical limit; Cauchy theory uniform in Knusen number, exponential rate of conergence towars global euilibrium Acnowlegements:This wor was supporte by the UK Engineering an Physical Sciences Research Council EPSRC grant EP/H23348/1 for the Uniersity of Cambrige Centre for Doctoral Training, the Cambrige Centre for Analysis Contents 1 Introuction 1 2 Main results 6 3 The linear part: a C -semigroup in spaces with polynomial weight, proof of Theorem An a priori estimate for the full perturbe euation: proof of Theorem 23 2 References 28 1 Introuction This article eals with the Boltzmann euation in a perturbatie setting as the Knusen number tens to zero This euation rules the ynamics of rarefie gas particles moing on the flat torus in imension, T, when the only interactions taen into account are binary collisions More precisely, the Boltzmann euation escribes the time eolution of the istribution f = ft, x, of particles in position x an elocity A formal eriation of the Boltzmann euation from Newton s laws uner the rarefie gas assumption can be foun in [14], while [15] presents 1

2 2 M BRIANT, S MERINO-ACEITUNO, C MOUHOT Lanfor s Theorem see [25] an [16] for etaile proofs which rigorously proes the eriation in short times We enote the Knusen number by an the Boltzmann euation reas t f + x f = 1 Qf, f, on T R, where Q is the Boltzmann collision operator gien by Qf, f = B, cos θ [f f ff ] σ R S 1 The Boltzmann ernel operator B encoes the physics of the collision process an f, f, f an f are the alues taen by f at,, an respectiely, where = + 2 = σ 2 σ 2, an cos θ =, σ The Boltzmann collision operator comes from a symmetric bilinear operator Qg, h efine by Qg, h = 1 B, cos θ [h g + h 2 g hg h g] σ R S 1 It is well-nown see [14], [15] or [17] for example that the global euilibria for the Boltzmann euation are the Maxwellians, which are gaussian ensity functions epening only on the ariable Without loss of generality we consier only the case of normalize Maxwellians: µ = 1 e 2 2π 2 2 In this paper we will assume that the Boltzmann collision ernel is of the following form 11 B, cos θ = Φ b cos θ, with Φ an b positie functions This hypothesis is satisfie for all physical moel an is more conenient to wor with but o not impee the generality of our results We also restrict ourseles to the case of har potential γ > or Maxwellian potential γ =, that is to say there is a constant C Φ > such that 12 Φz = C Φ z γ, γ [, 1], with a strong form of Gra s angular cutoff see [18], expresse here by the fact that we assume b to be C 1 with the controls from aboe 13 z [ 1, 1], bz, b z C b

3 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 3 11 The problem an its motiations The Knusen number is the inerse of the aerage number of collisions for each particle per unit of time Therefore, as reiewe in [34], one can expect a conergence, in some sense, from the Boltzmann moel towars the acoustics an the fluis ynamics as the Knusen number tens to Howeer, these ifferent moels escribe physical phenomena that o not eole at the same timescale an the right rescaling to approximate the incompressible Naier-Stoes euation see [4][17][34][31] is the following euation 14 t f + 1 xf = 1 2 Qf, f, on T R, uner the linearization f t, x, = µ + h t, x, This leas to the perturbe Boltzmann euation 15 t h + 1 xh = 1 2 Lh + 1 Qh, h, where we efine Lh = 2Qµ, h The hyroynamical limit of the perturbe euation is the system of euations satisfie by the limit, as tens to, of the hyroynamical fluctuations that are the following physical obserables of h : ρ t, x = h t, x,, R u t, x = h t, x,, R θ t, x = 1 2 h t, x, R Note that ρ, u, θ are the linearise fluctuations of the mass, momentum an the thermal energy aroun the global euilibrium µ In our perturbatie framewor, preious stuies [4][5][12] show that the hyroynamical limits ρ, u an θ are the wea in the Leray sense [26] solutions of the linearize incompressible Naier-Stoes euations: 16 t u ν u + u u + p =, u =, t θ κ θ + u θ =, where p is the pressure function an ν an κ are constants etermine by L see [4] or [17] Theorem 5 They also satisfy the Boussine relation 17 ρ + θ = The aim of the present wor is to use a constructie metho to obtain existence an exponential ecay for solutions to the perturbe Boltzmann euation 14, uniformly in the Knusen number One will thus be allowe to extract a conerging at least wealy subseuence of h conerging to the incompressible Naier-Stoes euations [5][4][12] Such uniform results hae been obtaine on the torus in Sobole spaces with exponential weight H x, in [23][12] an the present wor improes

4 4 M BRIANT, S MERINO-ACEITUNO, C MOUHOT this strong weight to a polynomial weight without the nee of eriaties in the elocity ariable 12 Existing results The first part of our wor is to proe that the linear part of the Boltzmann euation G = 1 2 L 1 x generates a strongly continuous semigroup with an exponential ecay in Lebesgue an Sobole spaces with polynomial weight, namely 1+ for some large enough It has been nown for long that the linear Boltzmann operator L is a self-ajoint non positie linear operator in the space L 2 µ Moreoer it has a spectral gap λ This has been proe in [13][18][19] with non constructie methos for har potential with cutoff an in [7][8] in the Maxwellian case These results were mae constructie in [2][27] for more general collision operators One can easily exten this spectral gap to Sobole spaces H see for instance [21] Section 41 The next step is to see if the latter properties about L in the elocity space can be transpose when one as the sew-symmetric transport operator x The first results were obtaine in [32] where G 1 was proen to generate a strong continuous semigroup in L 2 H x an in L Hx 1 +, for s an large enough Then [29] obtaine constructiely this result in H x, using hypocoerciity properties of the Boltzmann linear operator Finally, a recent breathrough proing abstract extension of semigroups [21] showe that G 1 generates a C -semigroup in all the Sobole spaces of the form W α, Wx β,p m, for m being an exponential weight incluing maxwellian ensity if = p = 2 or a polynomial weight 1 +, as long as α β an is large enough epening on with > 2 in the case = 1 The full Boltzmann euation perturbe aroun a global euilibrium µ 15 has also been stuie in the case = 1 The associate Cauchy problem has been wore on oer the past fifty years, starting with Gra [2], an it has been stuie in ifferent spaces, such as L 2 H x spaces [32] or Hx, 1 + [22][36] The Cauchy theory was then extene to H x, where an exponential tren to euilibrium has also been obtaine This was obtaine using hypocoerciity properties of the linear operator [29] or nonlinear estimates on flui an microscopic parts of the euation [23] Recently, [21] proe existence an uniueness for 15 in more the general spaces W α,1 W α, W x β,p 1 + for α β an β an large enough with explicit threshols This result therefore gets ri of the exponential weight neee in the preious stuies All the results presente aboe hol in the case of the torus We refer the reaer intereste in the Cauchy problem, both for the torus an the whole space, to the reiew [33] For physical purposes, these stuies for = 1 are releant since mere rescalings or changes of physical units changes 14 to the case where the Knusen number euals 1 Howeer, if one wants to stuy the hyroynamical limits of the Boltzmann euation, one nees to obtain explicit epenencies on the Knusen number Using hypocoerciity methos [12] gae a constructie uniform approach on the semigroup generate by G in H x, an its exponential ecay The stuy of the full perturbe Boltzmann euation 15 taing into account the epenencies on the

5 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 5 Knusen number has been obtaine [23][12] in the same spaces H x,, for s large enough More precisely, for initial ata sufficiently close to µ there exists a uniue non-negatie solution to 14 an it ecays exponentially fast towars its euilibrium The smallness assumption was proen to be inepenent of the Knusen number as well as the rate of ecay an the methos use in [12] are constructie 13 Our contributions an strategy The present wor brings two major improements In the spirit of [21], we first proe that G generates a strong continuous semigroup in Sobole spaces W α,1 W x β,p 1 + for α β an β an large enough with explicit threshols It is one by starting from existing results in H x, an then ecomposing the linear operator G into a issipatie part an a regularising part that is then treate in more an more regular spaces up to the space where the semigroup properties hae been erie in preious articles We thus improe the existing result [12] Our main contribution is an aapte ersion of the abstract extension theorem eelope in [21] that taes into account the epenencies on the Knusen number as well as a careful stuy of the issipatie an the regularising parts of the operator G The secon contribution of this article is the solution to the Cauchy problem with exponential tren to euilibrium, inepenently on, in spaces W α,1 Wx β, an W α,1 Hx β , for β large enough an all α β First, this result maes the recent stuy [21] uniform in the Knusen number Secon, it improes the Cauchy theory eelope uniformly in in [23][12] by ropping the exponential weight an the -eriaties Moreoer, one can notice that the polynomial weight is almost the optimal one for the Boltzmann euation conseration of mass an energy The main issue to obtain uniform results is that the bilinear operator 1 Q cannot be treate as a mere perturbation that eoles uner the flow of S G, the semigroup generate by G, since the latter has an exponential ecay of orer O1 that is negligeable compare to O 1 as tens to zero We eelop an analytic point of iew about the extension theorem in [21] an inclue the bilinear term More precisely, we ecompose the perturbe euation 15 into a hierarchy of euations taing place in spaces that hae more an more regularity up to H x, where estimates ha been erie in [12] At each step we use the issipatie part of the linear operator to control the remainer term 1 Q whereas the regularising part is controlle in spaces with higher regularity 14 Organization of the article Section 2 first introuces the ifferent notations an efinitions we will use throughout the paper an then states the precise theorems we proe in this wor Section 22 eals with the semigroup generate by the full linear operator 2 L 1 x whereas Section 23 is eicate to the full Boltzmann euation The full linear part of the Boltzmann operator is proen to generate a strongly continuous semigroup in Lebesgue an Sobole spaces with polynomial weight in Section 3

6 6 M BRIANT, S MERINO-ACEITUNO, C MOUHOT We start with Section 31, a thorough escription of our strategy an a ersion of the extension theorem of [21] that taes into account the epenencies in We show in this section that 2 L 1 x can be ecompose into a regularising operator in the elocity ariable Section 32 an a issipatie one Section 33 We then combine the last two properties to gain regularity both in space an elocity Section 34 to finally proe the existence an exponential ecay of the associate semigroup Section 35 The last section, Section 4, proes existence, uniueness an exponential ecay of solutions to the perturbe Boltzmann euation 15 Section 41 gies a new point of iew on the extension we use to generate the semigroup associate to 2 L 1 x an how it can be use with the bilinear operator This strategy is eelope through Sections 42 an 43 an it leas to the proof of the exponential ecay towars euilibrium in Section 44 2 Main results 21 Notations We gather here the notations we will use throughout this article Function spaces We first efine the following shorthan notation, = The conention we choose is to inex the space by the name of the concerne ariable so we hae, for p in [1, + ], L p [,T ] = Lp [, T ], L p x = L p T, L p = L p R Let p an be in [1, +, α an β in N an m : R R + a strictly positie measurable function For any multi-inexes j = j 1,, j an l = l 1,, l in N we enote the j, l th partial eriatie by We efine the space W α, j l = l x j Wx β,p m by the norm f W α, Wx β,p m = j α, l β l + j maxα,β where we use the Lebesgue norm j l f m L L p x, g L L p x = [ R T fx, p x /p ] 1/ Linear Boltzmann operator First we use a writing conention The present wor aims at extening results nown in a small space E, namely H x, with s sufficiently large, into a larger space E, namely Lebesgue an Sobole spaces with polynomial weight We will use curly letters for operators in E an their non-curly euialent to enote their restriction to E For instance, we will enote L E = L The linear Boltzmann operator L has seeral properties we will use throughout this paper see [14][15][35][21] for instance

7 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 7 L is a close self-ajoint operator in L 2 µ with ernel Ker L = Span {φ,, φ +1 } µ, where φ = 1, for i = 1,, we efine φ i = i an φ +1 = 2 / 2 The family φ i i +1 is an orthonormal basis of Ker L in L 2 µ an we enote π L the orthogonal projection onto Ker L in L 2 µ : π L h = huφ i u u φ i µ, R i= an we efine π L = I π L We will also enote the full linear Boltzmann operator by For s in N we will use the conention G = 1 2 L 1 x G H s x,µ = G It has been proen [12] Proposition 31 that the ernel of G oes not epen on an that its generators in L 2 x, µ are the same than the ones of Ker L We therefore hae that the orthogonal projection onto Ker G in L 2 x, µ is gien by Π G h = Π G h = hx, uφ i u xu φ i µ, T R i= an we efine Π G = I Π G Note that for a function h in L 2 x, µ we hae that x, T R, Π G hx, = π L hx, x T 22 Results about the full linear part We first eal with G, the linear part of the perturbe Boltzmann operator We proe that it generates a strongly continuous semigroup with an exponential ecay in Lebesgue an Sobole spaces with a weight as long as is large enough The precise statement is the following Theorem 21 Let B be a Boltzmann collision ernel satisfying There exists < 1 such that for all p, in [1, + ], all α, β in N with α β an all >, where 23 = / 2 with γ efine in 12, + γ for all <, G = 2 L 1 x generates a C -semigroup, S G t, on W α,, W β,p x

8 8 M BRIANT, S MERINO-ACEITUNO, C MOUHOT 2 for all τ >, there exist C G τ, λ >, such that for all < an for all h in in W α, W x β,p, for all t τ S G th in Π G h in W α, W β,p x C Gτe λt h in Π G h in W α, where Π G is the spectral projector onto Ker G which is gien, for all, by Π G g = gφ i x φ i µ T R i= W β,p x, The constants, C G τ an λ are constructie an only epens on, p,,, α, β an the ernel of the Boltzmann operator We refer to [24] an [21] Section 2 for efinitions an properties of spectral projectors Remar 22 We can mae a couple of remars about this theorem 1 Of important note is the fact that the exponential ecay has a strong epenence on τ > This is inherent to our uniform extension of the enlargement semigroup theory of [21] Howeer, such a problematic initial layer coul not be use to control irectly the bilinear operator That is why, as explaine in the introuction, we nee a new analytic approach of the enlargement theorem to tacle the non-linearity 2 It has been proen in [12] Section 3, that in H 1 x,µ, Ker G oes not epen on if is positie an we therefore can efine Π G = Π G During the proof of Theorem 21 we will show that Π G H s x,µ = Π G an thus Π G is well-efine an is inepenent of an gien by 22 3 As notice in [21], the rate of ecay λ can be taen eual to the spectral gap of L H s x,µ see [12], for s as large as wante, when is big enough an we obtaine a constructie threshol 4 Finally, we emphasize that in the case = 1, the result hols for all > 2 This is almost the minimal regularity L for the Boltzmann euation, that is solutions with boune mass an energy 23 Existence, uniueness an tren to euilibrium A physically releant reuirement for solutions to the Boltzmann euation are that their mass, momentum an energy are presere with time This is also an a priori property of the Boltzmann euation on the torus see [35] Chapter 1 Section 2 for instance which reas t, T R 1 2 f t, x, x = T R 1 2 f, x, x If one expects tren to the euilibrium µ for the solutions f = µ + h of the Boltzmann euation 14 then it must be that 1 t, h t, x, x =, T R 2

9 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 9 that is Π G h t,, = for all t, which is a property that is inee presere along time for solution to the perturbe Boltzmann euation 15, see [12] for instance We hence state the following theorem answering the Cauchy problem an the exponential conergence towars the euilibrium µ Theorem 23 Let B be a Boltzmann collision ernel satisfying an let p = 1 or p = 2 For α, β an > 2 in N efine E p = W α,1 W x β,p There exists < 1 an β in N such that for all β β, α β, for all > 2, for any λ in, λ λ efine in Theorem 21 there exist C α,β, η α,β > such that for any <, for any istribution f in = µ + h in : If i h in is in KerG in E p, ii h in E p η α,β, then there exists a uniue global solution f = f t, x, to 14 in E p which, moreoer, satisfies f = µ + h with: h belongs to KerG for all times, h E p C α,β h in E p e λ t The constants C α,β an η α,β are constructie an epens only on α, β,,, λ an the ernel of the Boltzmann operator Remar 24 Note that the uniform control in E p gies a wea conergence of the of the physical obserables of h t, x,, that is ρ t, x, u t, x an θ t, x Thans to the conergence in istribution stuy mae by Baros, Golse an Leermore [4][3] we now that the wea limit of ρ, u, θ is a solution to 16 with 17 close to the euilibrium 1,, 1 3 The linear part: a C -semigroup in spaces with polynomial weight, proof of Theorem 21 In this section we focus on the linear part of the perturbe Boltzmann euation We thus consier the following euation: in W α, W β,p x 31 t h = G h 31 Strategy of the proof If we enote E = W α, W x β,p an E = H x, we hae that E E, ense with continuous embeing for s large enough [12] Theorem 21 with the norm of Theorem 24 states that G = G E generates a strongly continuous semigroup in E with exponential ecay Theorem 21 can therefore be unerstoo as the possibility to exten properties of G in a small space E to G in a larger space E This issue of extening spectral gap properties as well as semigroup properties has been first tacle by Mouhot to obtain constructie rates of conergence to euilibrium for the homogeneous Boltzmann euation [28] Recently, Gualani,

10 1 M BRIANT, S MERINO-ACEITUNO, C MOUHOT Mischler an Mouhot [21] propose a more abstract approach that allows to eal with the full linear operator In their wor, they proe that if some conitions on G an G are satisfie then we can pass on some semigroup properties from E to E The main argument of the proof of Theorem 21 is to show that we can use their result in our setting, inepenently of To be more precise, we gie below a moifie ersion of their main functional analysis theorem which is combination of Theorem 213 an Lemma 217 where we ae epenencies on We refer to [21] Section 2 for the efinition of hypoissipatiity roughly speaing it is a issipatie property in a ifferent norm on a Banach space an the efinition of the conolution of two semigroups of operators enote by the symbol In the seuel we will use C E for the set of close operators on E an BE for the set of boune operators on E For any operator G in C E we enote RG its range an ΣG its spectrum Theorem 31 Moifie extension theorem from [21] Let be a parameter such that < 1 Let E, E be two Banach spaces with E E ense with continuous embeing, an consier G in C E, G in C E with G E = G an a > We assume the following A1 G generates a semigroup S G on E, G +a is hypoissipatie on R I Π G,a an Σ G {z C, Rez > a} = {} where is a iscrete eigenalue A2 There exists A, B in C E such that G = A + B with corresponing restrictions A, B on E an there exist some intermeiate spaces not necessarily orere E = E J, E J 1,, E 2, E 1 = E such that, still enoting B := B Ej an A := A Ej i For any j in {1,, J}, B + a/ 2 is hypoissipatie on E j ; ii For any j in {1,, J}, A B E j with A BEj C A/ 2 ; iii there are some constants l, l 1 N, C 1, K R, α [, 1 such that t, T l t BEj,E j+1 C ekt/2 l 1 t α, for 1 j J 1, with the notation T l := A S B l Then G is hypoissipatie in E an for all a < a there exists n = na 1 an some positie constants C a an C a such that 32 T n t BE C a a t/2 nl 1/l e ; 33 n 1 S G t = S G tπ G + 1 l I Π G S B T l t + 1 n [I Π G S G ] T n t; 34 l= BE S G t S G tπ G C a t n n2+l 1/l t/2 e a,

11 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 11 where Π G has been efine in 24 We will use Theorem 31 to irectly proe Theorem 21 Inee, [12] Theorem 21 states that G generates a strongly continuous semigroup with exponential ecay at rate a in E = H x,, which is the reuire assumption A1 properties about the spectral gap of the spectrum can be foun in [2] Therefore if G fulfils hypothesis A2 then it generates a strongly continuous semigroup, with an exponential ecay of orer λ for all < λ < a Inee, tae a in λ, a in 34 an fix τ > an see that 35 t τ, t n t/2 n2+l e a 1/l t n λ n2+l 1/l e a t/2 which is the expecte result gien in Theorem 21 e λ t/ 2 C n,l1 /l,τ,a,λ e λ t, 32 Decomposition of the operator an assumption A2ii In this section we fin a ecomposition G = A + B that will fit the reuirements A1 A2 of Theorem 31 This ecomposition has been foun in [21] in the case = 1 We will use exactly the same operators but incluing the epenencies in All the results presente in the rest of this section are true for = 1 see [21] Section 4 so we will try to relate as much as possible our computations with the ones for = 1 For δ in, 1, to be chosen later, we consier Θ δ = Θ δ,, σ in C that is boune by one on the set { δ 1 an 2δ δ 1 an cos θ 1 2δ } an whose support is inclue in { 2δ 1 an δ 2δ 1 an cos θ 1 δ } We efine the splitting with an where G = A δ + B δ, A δ h = 1 2 R S 1 Θ δ [µ h + µ h µh ] b cos θ γ σ B δ h = B 2,h δ 1 νh 1 2 xh, B δ 2,h = 1 2 R S 1 1 Θ δ [µ h + µ h µh ] b cos θ γ σ an ν is the stanar collision freuency ν = b cos θ γ µ σ R S 1 Note that there exists ν, ν 1 > such that 36 R, ν 1 + γ ν ν γ We hae that A δ = 1 2 Aδ 1 an B δ 2, = 1 2 Bδ 2,1

12 12 M BRIANT, S MERINO-ACEITUNO, C MOUHOT We therefore obtain the following controls on A δ Proposition 32 For all < <, for any in [1, + ] an α, the operator A δ maps L into W α, with compact support There exists C δ,α,, R δ > inepenent of such that h L, supp A δ h B, R δ, A δ h W α, C δ,α, h 2 L Moreoer, for any p in [1, + ] an for all h in L L p x, A δ h L A δ L p h L p x x L Remar 33 We notice here that this Proposition gies the point A2ii of Theorem 31 if the E j are Sobole spaces Proof of Proposition 32 The ernel of the operator A δ is of compact support so its Carleman representation see [13] gies the existence of δ in Cc R R such that 37 A δ h = 1 R δ, 2 h, an therefore the control on h is straightforwar The control of which states [ T A δ h L L p x R δ, hx, A δ W α, comes irectly from Minowsi s integral ineuality p ] 1/p x R 1/p δ, p hx, p x T 33 Dissipatiity estimates for B δ, assumption A2i One can fin in [21] proof of Lemma 414 case W 2 an W 3 the following estimate on the operator B δ in the case = 1 Lemma 34 For all p, in [1, + ], for all > 2 an for any δ in, 1 an all h in L L p x, h p sgnh h p 1 B δ L p R x 1 h x [ Λ γ/,δ 1 ] h T L L p x ν 1/, where is the conjugate exponent of an Λ, δ is a constructie constant such that 1/ 1 1/ 4 4 lim Λ,δ = φ = δ + 2 1

13 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 13 Remar 35 As notice in [21] Remar 43, the uantity φ is strictly less than one for bigger than a constant The constant we are consiering is not optimal an is such that φ γ/ < 1, where is the conjugate exponent of This appearance of γ/ is ue to a loss of weight of orer ν 1/ in the estimate of the spectral gap, see proof of Proposition 36 In the case of the Boltzmann operator with har potential an angular cutoff, point A2i is fulfille by B δ for δ small enough This is the purpose of the following lemma We recall here that ν = inf ν > an that we efine R W α, Wx β,p = j l L L p x l + j maxα,β j α, l β Proposition 36 Consier p, in [1, + ], >, efine by 23, an α, β in N such that α β Then there exists δ, in, 1 such that for all < δ δ, there exists λ = λ,, δ in, ν such that for all < 1, λ,, δ tens to λ, as δ goes to, λ, tens to ν when goes to +, B δ + λ / 2 is hypoissipatie in W α, W x β,p Proof of Proposition 36 Let h be in W α, W x β,p an consiert h to be a solution to the linear euation 38 t h = B δ h = B 2,h δ 1 νh 1 2 xh, with initial alue h Since the x-eriatie commutes with the euation we can consier only the case when β = α The proof is split into two parts First we proe Proposition 36 in the case α = an then we stuy the case with -eriaties Step 1: the case α = Tae p, in [1, + We recall that h L L p x = [ R 1 + T h p x /p ] 1/ Therefore we can compute t h L L p x = h 1 L L p x h p sgnh h p 1 B δ L p R x h x T Obsering that sgnh h p 1 x h x = 1 T p x h p x =, T

14 14 M BRIANT, S MERINO-ACEITUNO, C MOUHOT we euce t h L L p x = h L L p x R 1 + h p L p x We can therefore use Lemma 34 which leas to 31 sgnh h p 1 B δ 1 h x T t h L L p x 1 2 [ 1 Λ γ/,δ ] h L L p x ν 1/ h 1 L L p x, We alreay notice that Λ 1/,δ is strictly less than 1 for δ smaller than some δ, see Remar 35 Therefore, because ν ν for all we hae that for all δ smaller than δ, the following hols, t h L L p x ν 2 [ 1 Λ γ/,δ ] h L L p x, This conclues the proof of Proposition 36 for α = an 1 p, < + The cases p = an = are respectiely ealt with by taing the limit p an which is possible since δ, is inepenent of p an can be chosen to conerge to a strictly positie constant when goes to, thans to the efinition of Λ, δ Step 2: the case with -eriaties Tae p, in [1, + ] an α = β = 1 Since the x-eriatie commutes with 38 the euation satisfie by h, we hae that 31 hols for x-eriaties Notice that 1 gies 311 t h L L p x + xh L L p x ν1 1/ [ 1 Λ γ/ 2,δ ] h L L p x ν 1/ + xh L L p x ν 1/ Applying a -eriaties to 38 yiels t h = B δ h + B δ h = B δ h 1 xh + R δ h, where R δ h = B δ 2, h 1 2 νh = 1 R δ 2 1 h From 311, our computations in Step 1 with δ δ, an the following norm h W 1, W 1,p x η = h L L p x + xh L L p x + η h L L p x, with η > to be fixe later, we obtain

15 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 15 t h W 1, Wx 1,p η ν1 1/ [ 1 Λ γ/ 2,δ ] h L L p x ν 1/ + xh L L p x ν 1/ η ν1 1/ [ 1 Λ γ/ 2,δ ] h L L p x ν 1/ η h 1 L L p x + η 2 h 1 L L p x h p L p R x h p L p R x sgnh h p 1 x h x T sgnh h p 1 R δ 1 h x T We tae the absolute alue an use Höler ineuality twice on the last two terms which maes the terms in h isappear, an this gies t h W 1, Wx 1,p η ν1 1/ 2 [ 1 Λ γ/,δ ] h L L p x ν 1/ + η h L L p x ν 1/ + 1 ην 1/ 2 ν 1 1/ [ 1 Λ γ/,δ ] x h L L p x ν 1/ + η R δ 2 1 h L L p x One can fin in [21] proof of Lemma 414 case W 2 an W 3 the following estimate R δ 1 h L L p x C δ h L L p x ν 1/, where C δ > is a constant only epening on δ Because 1, this latter estimates yiels 312 t h W 1, Wx 1,p 1 C η 2 δ η ν 1 1/ + 1 ην 1/ 2 ν 1 1/ [ 1 Λ γ/,δ ] h L L p x ν 1/ [ 1 Λ γ/,δ ] x h L L p x ν 1/ η ν1 1/ [ 1 Λ γ/ 2,δ ] h L L p x ν 1/, which conclues the proof if we tae η small enough in terms of δ, for δ δ, The case where 1 < α = β is ealt with in the same way with the norm h W α, Wx α,p = η j j l h η L L p x, j + l α with η small enough in terms of δ

16 16 M BRIANT, S MERINO-ACEITUNO, C MOUHOT 34 Estimates on the iterate conolution prouct, assumption A2iii In orer to use Theorem 31, it remains to show that our euation 31 satisfies hypothesis A2iii, that is we nee to control the iterate uantities T l := A δ S B δ when p = 1 l for some l in N The following proposition escribes such controls Proposition 37 Consier >, efine by 23, an s in N For any δ in, δ, ] an any λ in, λ δ, an λ efine in Proposition 36, there exists C 1 = C 1 λ, δ > an R = Rδ > such that for any t, an 313 s 1, n N, supp T n th K := B, R T 1 th W s+1,1 x, K C 1 e λ 2 t 2 t e λ 2 t h W s+1,1 W s x, 314 s, T 2 th C s+1/2,1 W x, K 1 h 4 W s,1 x, Proof of Proposition 37 Most of the proof is an aaptation of [21] proof of Lemma 419 to eep trac of the epenencies on We will refer to it when we are using some of its computations Control of T 1 th: The x-eriaties commutes with T 1 t an therefore it is enough to consier h in W s,1 Wx 1,1, with s 1, an to control T 1 th W s+1,1 Wx 1,1 K This yiels 315 T 1 th W s+1,1 Wx 1,1 K T 1th W s+1,1 L 1 x K + xt 1 th W s+1,1 L 1 xk The first term is easily ealt with thans to the estimate on A δ, Proposition 32, an the issipatiity property of B δ, Proposition 36, 316 T 1 th W s+1,1 A L 1 xk = δ S δ B h W s+1,1 For the secon term, efine ft = S δ B th an 317 D t = 1 t x + By irect computations we hae that 1 t x T 1 th = A δ C λ L 1 x K 2 e D t f A δ f 2 t h L 1 L 1 x Aboe we enote A δ to be the operator A δ where the ernel, is swappe with,, see the proof of Proposition 32 By Proposition 32 we get t x T 1 th W s+1,1 L 1 xk C ] [ D 2 t f L 1 x, + f L 1 x, The issipatiity property of B δ, in particular 31 with = 1, yiels

17 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES t f L 1 x, λ f 2 L 1 x, ν Direct computations yiels where t D t f = B δ D t f J δ f, 32 J δ = B δ B δ with B δ = B δ 2,1 ν is inepenent of an satisfies see [21] proof of Lemma 419 for all g in L 1 J δ g L 1 C δ g L 1 ν In the same way as proof of Proposition 36 we obtain ν 321 t D tf L 1 x, λ D tf 2 L 1 x, ν + C δ f 2 L 1 ν We then consier λ in, λ an efine η = λ λ /C δ We compute, with 319, [ e λ t 2 t η D t f L 1 x, + f L 1 x, ], an thus 322 D t f L 1 x, + f L 1 x, η 1 e λ 2 t h W 1,1 L 1 x To conclue we plug 322 into 318 an we combine it with 316 into 315 This yiels, because s 1, T 1 th W s+1,1 Wx 1,1 K λ e C 2 t 2 t h W s,1 L 1 x, which implies the expecte result 313 because T 1 t commutes with x-eriaties Control of T 2 th: For s we can interpolate for interpolation theory in Sobole spaces see [6] Chapters 6 between 316 an 313 to get T 1 th s+1/2,1 W x, K λ e C 2 t 2 t h W s,1 Then, we firstly use the ineuality aboe an seconly 316 to obtain L 1 x T 2 th W s+1/2,1 x, K t T 1 t st 1 sh s s+1/2,1 W x, K C λ t 4 e t 2 e λ λ 2 s s h t s W s,1 L 1 x, which is the expecte result 314

18 18 M BRIANT, S MERINO-ACEITUNO, C MOUHOT The aim is to lin our space L L p x to the space H x, We thus state the following control on the iterate conolution in the case p = 2 Proposition 38 Consier >, efine by 23, an s in N For any δ in, δ, ] there exists C 2 = C 2 δ > an R = Rδ > such that for any t, n N, supp T n th K := B, R an 323 s, T 2 th C T s+1/2 H x, K h 5/2 Hx, s Proof of Proposition 38 Consier h in Wx, s,2, s in N This Proposition is easier than when p = 1 because there exists elocity aeraging lemmas in this framewor, as iscusse in [21] Remar 421 The x-eriatie commutes with T 1 an therefore we suppose there is no eriatie in space Define ft = S δ B th so that f is solution to the inetic euation t f + 1 xf = s t, x,, with s t, x, = 2 νf + 2 B 2,1f δ Let j be a multi-inex such that j s We apply j to the latter euation, which gies 324 t j f + 1 x j f = j s t, x, + 1 i + l = j a i,l i l f, where a i,j are non-negatie numbers A classical aeraging lemma see [9] Lemma 1 an [1] in which we emphasize the epenencies in reas, for 324 with j f, x, = j hx,, for all ψ in D R 325 ft, j x, ψ R C L 2 t Hx 1/2 j hx, L 2 x, + j s L 2 t,x, + 1 i + l = j a i,l l i f L 2 t,x, We use [21], Lemmas 44 an 47, in orer to boun the terms inoling B δ 2, = 2 B δ 2,1 we hae that s H s x, 1 2 s 1 H s x, C 2 f H s x, ν C 2 e λ 2 t h H s x, ν, where the last ineuality comes from the hypoissipatiity properties of S B t, see Proposition 36 Using the issipatiity properties of S B t one more time we euce that

19 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES T 1 th L 2 t H s+1/2 x, C 5/2 h H s x, ν To conclue we notice that t T 1t st 1 s s is a continuous linear operator on the Hilbert space Hx, s+1/2 K an thus we can see it as an element of Hx, s+1/2 K by Riesz s representation theorem Hence, thans to Cauchy-Schwartz, T 2 th H s+1/2 x, K = t T 1 t st 1 s s t h H s x, ν t h H s x, ν h H s x, ν h H s+1/2 x, K T 1 t st 1 s B s H x, s ν,hx, s+1/2 K T 1 t s 2 B H s x,k,h s+1/2 x, K t 1/2 T 1 s 2 BHx, s ν,hx, s K s C 5/2 h H s x, ν, C t C A 5/2 λ 2 e where we use Proposition 32 an the fact that S δ B Hx, s with spectral gap λ / 2 2 s s 1/2 s 1/2 is a contraction semigroup on 35 Proof of Theorem 21 As we explaine it in Section 31, the proof of Theorem 21 is irect from the application of Theorem 31 This theorem is clearly applicable in our case an we emphasize it through the extreme case of no eriatie in space or elocity ariables Inee, we consier s in N to be chosen big enough later We efine E = L L p x an E = H x, an we hae E E for s big enough ense with continuous embeing Inee, in the case 2 an p 2, stanar Sobole embeings see [11] Section IX3 imply E L L p x µ In the case p < 2 we hae, on the torus, L 2 x L p x an Hx s L 2 x by the same Sobole embeings Finally, in the case < 2 we hae that L 2 µ L it can be one by a mere Cauchy-Schwarz ineuality an the same Sobole embeings gie H L 2 µ On the torus we hae the following embeing: L p x L 1 x Thans to Proposition 32 an Proposition 36 we obtain same arguments as T 1 th E C A δ S B δ We therefore efine E 2 = L 1 L 1 x h C λ L L 1 x K 2 e 2 t h L 1 L 1 x

20 2 M BRIANT, S MERINO-ACEITUNO, C MOUHOT Then we efine by E j = W x, j 2/2,1 for j from 2 to m with m big enough such that W x, m 1/2,1 L 2 x, Then we enote E j = H x, j m 1/2 up J 1 where H x, J m 2/2 E Point A1 of Theorem 31 is satisfie thans to [2] an [12] Theorem 21 with the norm of Theorem 24, point A2i by Proposition 36 an point A2ii by Proposition 32 Finally, point A2iii is gien by 327 for E an E 1, then by Proposition up to E m an by Proposition 38 from E m to E J an E 4 An a priori estimate for the full perturbe euation: proof of Theorem 23 In this section we wor in W α,1 H x β or in W α,1 W x β,1, with α β on the full perturbe Boltzmann euation t h = G h + 1 Qh, h 41 Description of the problem an notations When = 1, the linear part G has the same orer of magnitue than the bilinear term Q in the linearize Boltzmann euation 15 In this case, Theorem 21 suffices to obtain existence an exponential ecay since the contraction property of the semigroup S G1 controls the bilinear part for small initial ata see [21] In the general case, S G only generates a semigroup with a spectral gap of orer 1, insufficient to control 1 Q Howeer, [23][12] show that a careful stuy of 1 Q compare to G yiels existence an exponential ecay of solutions to 15 in H x, for s large enough see Theorem 47 for an aapte ersion of this result Our strategy is to use the same in of ieas as when we extene the semigroup properties from H x, β µ to W α,1 H x β an W α,1 W x β,1 but incluing the bilinear term Namely, we shall ecompose the partial ifferential euation 15 into a system of partial ifferential euations from W α,1 H x β or W α,1 W x β,1 to H β µ an use the perturbatie estimates of [12] As notice in Remar 216 of [21], Theorem 31 extening the semigroup generate by G in H to L 1 L x can be interprete as a ecomposition of t f = G f, into a system of partial ifferential euations, inoling operators G = A + B efine in Section 32, with f = f f J satisfying f 1 is in L 1 L x an fin 1 = f in in KerG, for all 2 j J 1, f j is in E j an f j in =, f J is in H, fin J = an in that space we can use the contraction property of S G We will ecompose the linearize Boltzmann euation in a similar way than the one explaine aboe We shall efine a seuence of spaces E j 1 j J In each space E j, 1 j J 1, a piece of the bilinear term, of orer 1, will be ae an controlle by the issipatiity property of B δ, of orer 2 Contrary to the stuy in the linear case, the bilinear operator generates terms inoling functions in all the spaces E j which hae to be compare an controlle This imposes to construct E j 1 j J as a neste seuence

21 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 21 The ifficult part of the linear operator, namely A δ, enjoys a regularising effect an coul therefore be treate in more regular spaces Of course, our ecomposition will be much easier since we solely want to go from an exponential weight into a polynomial weight Sobole spaces, without losing any eriaties in x or In orer to shorten notations we efine, for p = 1, 2 an to be efine later, 41 E p = W α,1 W x β,p an E = H x, β µ We tae h in in E p an we ecompose the partial ifferential euation, t h = G h + 1 Qh, h = Aδ h + B δ h + 1 Qh, h into an euialent system of partial ifferential euations for the following ecomposition 42 ht, x, = h t, x, + h 1 t, x,, with 1 In E p, h t= = h in an 43 t h = B δ h + 1 Qh, h + 2 Q h, h 1, 2 In E, h 1 t= = an 44 t h 1 = G h Qh1, h 1 + A δ h The aim of this Section is to establish the following estimate of solutions to the system Theorem 41 Let p = 1 or p = 2 There exist β in N an in, 1] epening on an the ernel of the Boltzmann operator such that: For all β β, for any δ in, δ,1 ] an any λ in, λ δ,1 an λ efine in Proposition 36 there exist C β, η β > such that for any < an h in in E p, if i h in E p η β, ii h, h 1 is solution to the system 43 44, then h + h 1 Ep C β h in Ep e λ t The constants C β an η β are constructie an epens only on β,, δ, λ an the ernel of the Boltzmann operator Remar 42 Lin with Theorem 23 The existence an uniueness for the perturbe Boltzmann euation 15 in E p has been proe for = 1, that is euialent of fixe with constant epening on it, in [21] Theorems 53 an 55 respectiely The constants, as well as the smallness assumption on the initial ata, in the theorem aboe are inepenent of an therefore this a priori result combine with existence an uniueness eelope in [21] an in [12] implies the existence an uniueness inepenently of which is Theorem 23

22 22 M BRIANT, S MERINO-ACEITUNO, C MOUHOT The next subsections eal with the estimates one can get for solutions to the system We stuy each of them inepenently an the a priori exponential ecay will be a straightforwar application of these results together with a maximum principle argument Section 42 focuses on the a priori stuy of the euation in E p Section 43 eals with 44 in E Finally, Section 44 gathers the preious results to proe Theorem Stuy of euation 43 in E In this section we proe the following general proposition about the euation taing place in E p = W α,1 W x β,p, for p = 1 or p = 2 We efine the shorthan notation Eν p = W α,1 Wx β,p ν Proposition 43 Let p = 1 or p = 2 an < 1 Let > 1 = 2, β > 2/p Let h in be in E p an h 1 in E p ν For any δ in, δ,1 ] an any λ in, λ δ,1 an λ efine in Proposition 36 there exist C, η > such that if i h in E p η, h 1 E p ν η, ii h satisfies h t= = h in an is solution to then t h = B δ h + 1 Qh, h + 2 Q h, h 1, h E p C e λ 2 t h in E p The constant η is constructie an epens only on δ, λ an the ernel of the Boltzmann operator Note that the presence of C is ue to the hypoissipatiity of the linear operator rather than a irect issipatiity We nee to control the bilinear term Q, which is gien by the following lemma Lemma 44 For all p = 1, 2 an α, β in N such that β > 2/p, there exists C β,p > such that all f an g Qf, g E p C β,p g E p ν f E p + g E p f E p ν This lemma has been proe in Lemma 516 in [21], which is aapte from interpolation results in [1] or uality arguments as in [3] Theorem 21 Proof of Proposition 43 Consier δ in, δ,1 ] an λ in, λ p = 2 an β > 2/p We hae that Tae p = 1 or t h = B δ h + 1 Qh, h + 2 Q h, h 1

23 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 23 Thans to the hypoissipatiity of property of B δ, more precisely the proof of Lemma 36, we hae h t E p λ λ 2 ν h E p ν + 1 h 2 ν E p ν + 1 Qh, h + 2Qh, h 1, h E p Qh, h + 2Qh, h 1 E p, where we use the scalar prouct notation to refer to the prouct operator appearing in W α,1 Wx β,p when one ifferentiates h W α,1 of the same form as 39 Wx β,p For the secon ineuality we use Höler ineuality between L p x an L p/p 1 x the prouct operator: sgnh h p 1 F h x h p 1 T L p x F h L p x insie Then estimating Q using Lemma 44 yiels 45 h t E p 1 [ λ h 2C 2 β,p ν Ep + 2 ] h 1 h ν, E p ν E p ν we recall ν = inf R ν > Therefore, if then h E p h 1 E p ν 1 λ λ 8C β,p an h t= E p 1 λ λ 4ν C β,p, is always ecreasing in time with h t E p λ h 2 ν E p ν, which hence yiels the expecte exponential ecay by Grönwall Lemma 43 Stuy of euations 44 in E In the space E = H x, β µ, solutions to the perturbe Boltzmann euation enjoy an exponential ecay More precisely, [12] erie a precise Grönwall that we will now use to obtain estimates on the solution h 1 We will use the following shorthan notation E ν = Hx, β µ ν 1/2 In this section we use the preious theorem to obtain exponential ecay of h 1 in E This result is state in the following proposition, where C t enotes the space of time-continuous functions

24 24 M BRIANT, S MERINO-ACEITUNO, C MOUHOT Proposition 45 Let p = 1 or p = 2, < 1, β s an α β an s being constructie constants that will be efine in Theorem 47 Let h in be in E p an h in Ct E p For any δ in, δ,1 ] an any λ in, λ δ,1 an λ efine in Proposition 36 there exist η 1, C 1 > such that if i h in E p η 1, ii there exists C > such that h E p C e λ +λ 2 2 t h in E p, iii h 1 satisfies h 1 t= = an is solution to t h 1 = G h Qh1, h 1 + A δ h then h 1 E C 1 e λ t h in E p The constants C 1 an η 1 are constructie an epens only on δ, λ an the ernel of the Boltzmann operator In orer to proe Proposition 45 we nee a new control on the bilinear term For any operator F efine on E E, we will say that F satisfies the property H if the following hols Property H: 1 for all g 1, g 2 in E we hae π L F g 1, g 2 =, where π L is the orthogonal projection on Ker L in L 2 µ see 21, 2 for all s > there exists FF s : E E R+ such that for all multi-inexes j an l such that j + l s, j l F g1, g 2, g 3 L 2 x,µ FF s g 1, g 2 g 3 L 2 x,µ ν 1/2, with F s F F s +1 F Lemma 46 The Boltzmann linear operator Q satisfies the property H with s >, C s >, F s Qf, g C s [ f E g Eν + f Eν g E ] The latter control on the bilinear part is from [12] Appenix A2 Proof of Proposition 45 We state below the estimate erie in [12] note that this is a ersion of [12] Theorem 24 extene by estimates proe in [12] Propositions 22 an 71 Theorem 47 There exist < 1 an s in N such that for any s s an any λ in, λ there exists δ s, C s > such that, for any h in in H x, with h in H s x,µ δ s, for any operator F efine on H x, H x, satisfying the property H;

25 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 25 Then for all < an for all g 1, g 2 in H x,, if h is a solution to t h = G h + 1 F g1, g 2 h t= = h in, an h is in Ker G for all time, then t R +, t h 2 H s x,µ 2λ h 2 ν 2 Hx,µ s ν + C s F s F g 1, g 2 2 Remar 48 It is of importance to notice that the ifferential ineuality gien in Theorem 47 is inepenent of In fact, it hols for a moifie H s x,, that inclues factors but in a way that H s x,, H inepenently of as long as x, s is sufficiently small For clarity purposes, woring irectly with the H s x, oes not hie any problematic behaiour If one wants to incorporate the real H s x,,, one just hae o exactly the following computations but with the integrate in time ersion of Theorem 47 where one can use the usual H s x, thans to the euialence of the norms Now, let λ be in, λ, s s an < The proof of Proposition 45 will be one in two steps First we stuy the projection of h 1 onto Ker G an then its orthogonal part Estimate on the projection part We hae that, see the ecomposition 42, that h 1 = h h with h solution to the perturbe Boltzmann euation an thus satisfying Π G h = We therefore hae that Π G h 1 = Π G h Moreoer, Theorem 21 tells us that Π G an Π G coincie on E an thus Π G h 1 = Π G h, an assumption ii together with the shape of Π G see 24, there exists a constant C Π >, epening only on the imension an s an the constant C, such that 46 Π G h 1 Eν C Π e λ +λ 2 2 t h in E p Estimate on the orthogonal part Applying Π G = I Π G, the orthogonal projection onto Ker G in L 2 x, µ, to the ifferential euation satisfie by h 1 yiels t Π G h 1 1 = G h 1 + Π G Qh1, h 1 + A δ h = G Π G h Π G Qh1, h 1 + A δ h Moreoer, we hae by efinition of Π G an π L see 24 an 21 that π L h = = Π G h =

26 26 M BRIANT, S MERINO-ACEITUNO, C MOUHOT an therefore Π G Qh 1, h 1 = Qh 1, h 1, since Q satisfies property H1 by Lemma 46 Plugging the latter euality into 47 gies t Π G h 1 = G Π G h Qh1, h 1 + Π G A δ h By efinition, Π G h1 is in Ker G for all time an thans to the control on the Boltzmann operator Q in E Lemma 46, we are able to use Theorem 47 with λ > λ to which we hae to a the source term Π G A δ h This yiels the following ifferential ineuality, where we enote by C any positie constant inepenent of, Π 48 t G h 1 2 E 2λ Π ν Gh E ν + C FQh s 1, h Π G A δ h, Π Gh 1 E 2λ Π ν Gh E ν + C h 1 2 h 1 2 E E ν + Π G A δ h E Π Gh 1 E, where we use a Cauchy-Schwarz ineuality on the last term on the right-han sie Then we can ecompose h 1 = Π G h 1 + Π G h1 to get first h 1 2 E h 1 2 E ν 4 Π G h 1 2 E + 4 ν 2 ΠG h 1 4 E ν, Π G h 1 2 E ν + 8 ν 2 Π G h 1 2 E ν Π Gh 1 2 E ν into which we can plug the control on Π G h 1 2 E ν we erie in 46 to obtain, with h in η 1, 49 h 1 2 E h 1 2 E ν 4 Π G h 1 2 E Π G h 1 2 E ν +Cη1 2 Π G h 1 2 E ν +Ce 2λ +λ 2 t h in 4 E p An finally, this ineuality together with assumption ii gies the existence of a constant C A > such that 41 Π G A δ h E Π G h 1 E C A 2 h in E p e λ +λ 2 2 t Π G h 1 E We plug 49 an 41 into 48 an obtain, with C an C being positie constants inepenent of, Π t G h 1 [ 2 2λ 4 E Π ν 2 G h 1 ] 2 + E Cη2 1 Π Gh 1 2 E ν + C h in 4 E p h in E p We now choose η 1 sufficiently small so that Cη1 2 λ λ, ν 2 Π G h 1 E e λ +λ 2 2 t

27 FROM BOLTZMANN TO INCOMPRESSIBLE NAVIER-STOKES IN SOBOLEV SPACES 27 which in turns implies 411 Π t G h 1 [ 2 λ + λ E ν 2 4 Π G h 1 2 E + C h in 4 E p h in E p ] Π Gh 1 2 E ν Π Gh 1 E e λ +λ 2 2 t We efine We hae that h 1 t= = so we can efine t = sup{t >, η = λ λ 4ν 2 Π G h 1 2 E < η } Suppose that t < +, we therefore hae for all t in [, t ] Π t G h 1 2 E 2λ Π ν 2 G h 1 2 E ν + C h in 4 η E + h p 2 in E p e λ +λ 2 2 t, which gies t [, t ], Π t G h 1 2 E 2λ Π G h 1 h 2 E +C in 4 η E + h p 2 in E p e λ +λ 2 2 t, an by Gronwall lemma with Π G h1 t= =, t [, t ], Π Gh 1 t h 2 E C in 4 η E + h p 2 in E p C 2 h in 4 E p + η h in E p + e λ +λ 2 2 s e 2λ s s e 2λ t e λ λ 2 u u e 2λ t, where we use the change of ariable u = 2 s an we consiere 1/4 which only amounts to ecreasing Hence, there exists K > inepenent of such that t [, t ], Π G h 1 2 E Kη4 1 + η 1 η If we thus chose η 1 sufficiently small such that η η 1 η K < η /2 we reach a contraiction when t goes to t since Π G h 1 2 E t η Therefore, choosing η 1 small enough inepenently on implies first that t = + an secon that 412 t [, +, Π G h 1 2 E C h in 2 E p e 2λ t En of the proof By just ecomposing h 1 into its projection an orthogonal part an using the estimates 46 an 412 gies the expecte exponential ecay for h 1 in E

28 28 M BRIANT, S MERINO-ACEITUNO, C MOUHOT 44 Proof of Theorem 41 Let p = 1 or p = 2, λ be in, λ, β β = s an < All the constants use in this section are the ones constructe in Proposition 43 with λ + λ /2 an Proposition 45 with λ E is continuously embee in Eν p because L 2 µ L 2 mere Cauchy- Schwarz ineuality an L 2 x L 1 x because T is boune Hence, there exists C E,E > such that ν E p E p ν C E,E E We efine η η = min η, η 1,, 2C E,E C 1 an we assume h in E p η Since h 1 t= = we also efine t = sup{t >, h 1 E p ν < η } Suppose that t < + Then, thans to Proposition 43 we hae that t [, t ], h E p h in E p e λ +λ 2 2 t We can thus apply Proposition 45 an get t [, t ], h 1 E C 1 h in E p e λ t C 1 η η, 2C E,E which is in contraiction with the efinition of t thans to 413 Therefore t = + an we hae the expecte exponential ecay state in Theorem 41 for all time References [1] Arery, L Stability in L 1 for the spatially homogeneous Boltzmann euation Arch Rational Mech Anal 13, , [2] Baranger, C, an Mouhot, C Explicit spectral gap estimates for the linearize Boltzmann an Lanau operators with har potentials Re Mat Iberoamericana 21, 3 25, [3] Baros, C, Golse, F, an Leermore, C D Flui ynamic limits of inetic euations II Conergence proofs for the Boltzmann euation Comm Pure Appl Math 46, , [4] Baros, C, Golse, F, an Leermore, D Flui ynamic limits of inetic euations I Formal eriations J Statist Phys 63, , [5] Baros, C, an Uai, S The classical incompressible Naier-Stoes limit of the Boltzmann euation Math Moels Methos Appl Sci 1, , [6] Bergh, J, an Löfström, J Interpolation spaces An introuction Springer-Verlag, Berlin- New Yor, 1976 Grunlehren er Mathematischen Wissenschaften, No 223 [7] Bobylë, A V The metho of the Fourier transform in the theory of the Boltzmann euation for Maxwell molecules Dol Aa Nau SSSR 225, , [8] Bobylë, A V The theory of the nonlinear spatially uniform Boltzmann euation for Maxwell molecules In Mathematical physics reiews, Vol 7, ol 7 of Soiet Sci Re Sect C Math Phys Re Harwoo Acaemic Publ, Chur, 1988, pp [9] Bouchut, F, an Desillettes, L Aeraging lemmas without time Fourier transform an application to iscretize inetic euations Proc Roy Soc Einburgh Sect A 129, , [1] Bouin, L, an Desillettes, L On the singularities of the global small solutions of the full Boltzmann euation Monatsh Math 131, 2 2, 91 18