Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules

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1 Ann. I. H. Poincaré A Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules G. Furioli a,, A. Pulvirenti b, E. Terraneo c,g.toscani b a University of Bergamo, viale Marconi 5, 0 Dalmine, Italy b Department of Mathematics, University of Pavia, via Ferrata, 700 Pavia, Italy c Department of Mathematics, University of Milano, via Saldini 50, 033 Milano, Italy Received July 009; received in revised form October 009; accepted 8 October 009 Available online 0 ovember 009 Abstract We prove propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for any value of the coefficient of restitution. The result follows from the uniform in time control of the tails of the Fourier transform of the solution, normalized in order to have constant energy. By standard arguments this implies the convergence of the scaled solution towards the stationary state in Sobolev and L norms in the case of regular initial data as well as the convergence of the original solution to the corresponding self-similar cooling state. In the case of weak inelasticity, similar results have been established by Carlen, Carrillo and Carvalho 009 in [] via a precise control of the growth of the Fisher information. 009 Elsevier Masson SAS. All rights reserved. Keywords: Granular gases; Kinetic models; Boltzmann equation. Introduction This paper concerns the regularity properties of solutions to the spatially homogeneous Boltzmann equation for Maxwellian molecules in with inelastic collisions, introduced in [6]. This equation describes the relaxation towards equilibrium of the distribution function of particles interacting through inelastic binary collisions. Kinetic theory of granular gases becames popular in the last ten years, and various mathematical problems, arising from dissipation, have been considered so far []. Let fv,τbe the probability density for the velocity v of a particle chosen randomly from the collection at time τ.letϕv be any bounded and continuous function on. For a dilute gas, with a mean free path of size ε, the equation under investigation is given, in weak form, by d dτ f,ϕ = Qf, f, ϕ ε * Corresponding author. addresses: giulia.furioli@unibg.it G. Furioli, ada.pulvirenti@unipv.it A. Pulvirenti, Elide.Terraneo@mat.unimi.it E. Terraneo, giuseppe.toscani@unipv.it G. Toscani. 09-9/$ see front matter 009 Elsevier Masson SAS. All rights reserved. doi:0.06/j.anihpc

2 70 G. Furioli et al. / Ann. I. H. Poincaré A where f,ϕ = ϕvf v,τdv, and Qf, f, ϕ = π S [ fv,τfw,τ ϕv ϕv ] dσ dv dw. In σ is a unit vector in S,dσ is the uniform measure on S with total mass π and the post-collision velocities are given by the collision mechanism written as v = e v + w + q + + e q σ, w = e v + w q + e q σ 3 with q = v w. The positive parameter e, with 0 e< is the restitution coefficient. From 3 it follows that the post-collision relative velocity is non-increasing, with v w = q = q e q q q σ q = v w. ote that the dissipation increases with e decreasing. Thus the case e = 0 corresponds to the strongest dissipation. Since e<, the collisions are inelastic, energy is dissipated in each collision, and the collisions are not reversible. This makes a crucial difference with the elastic theory in which there is a complete time reversal symmetry between the pre- and post-collision velocities. As pointed out in [], it is mainly for this reason that the Boltzmann equation is usually written in the weak form, and not because of any difficulty in constructing strong solutions. The postcollision velocities 3 represent one of the possible parameterizations of the inelastic collision mechanism. However, as exhaustively discussed in [] other possible pairs of pre-collision velocities v,w that result in the pair of post-collision velocities v, w can be constructed [6,9]. It is remarkable that these parameterizations give equivalent collision terms only when the restitution coefficient satisfies e>0 []. Consequently, any result which is valid for any value of e, including the case e = 0, cannot be translated, in this limit case, to other collision rules. Since the total momentum v + w is conserved in each individual collision v, w v,w, the first moment of fv,τi.e. the mean velocity is conserved. In particular, choosing as initial datum a probability density f 0 satisfying f 0 v dv =, v i f 0 v dv = 0, i =,, 3, f 0 v dv = T 0 <, it follows that while both mass and momentum are preserved in time, the second moment i.e. the temperature or energy decays according to the law Tτ= fv,τdv = T 0 exp { } e τ. 5 ε This implies that fv,τdv converges towards a point mass at v = 0asτ tends to infinity. The precise way in which the density collapses into a mass concentrated in zero has been investigated in various previous works [5,8,,7,9]. It has been shown that, after a certain relaxation time, each solution converges towards a self-similar solution, known as the homogeneous cooling state. The argument to show that this happens, and the respective rate of convergence, is based on an argument which is commonly used in nonlinear diffusion equations [5]. If one defines a temperature invariant scaling hv, τ of fv,τas Tτ 3/ Tτ / hv, τ = f v,τ, so that hv, τ dv = 3 for all τ 0, then the scaled density tends to a universal equilibrium state h, and h v dv = 3.

3 G. Furioli et al. / Ann. I. H. Poincaré A The equation for hv, τ now reads d dτ h, ϕ = e Qh, h, ϕ + h, v ϕ. 7 ε 8ε The speed at which the second moment converges to zero depends both on the mean free path ε and on the coefficient of restitution e. On the other hand, the dependence on the mean free path can be absorbed once and for all by scaling the time. By setting E = 8 e, 8 and scaling the time as t = e τ, 8ε we obtain for gv,t = h v, 8ε e t the equivalent equation d dt g,ϕ =E Qg,g,ϕ + g,v ϕ. In the rest of the paper, we will study Eq.. We remark that any result on the asymptotic convergence of gv,t towards g = h can be easily translated into a result on the asymptotic convergence of hv, τ. It is worth remarking that thanks to 6 and 0, the initial data g 0 of a scaled solution gt is related to the initial data f 0 of the original solution fτby 3/ / T0 T0 g 0 v = f 0 v 3 3 and so g 0 v dv = 3 independently of the initial temperature T 0 of f 0. Both the large-time behavior and the regularity of the solution of Eq. have been recently studied in []. In particular, propagation of regularity was found by controlling the growth of the frequencies of the Fourier transform of the solution by means of a precise control of the growth of the Fisher information I f t = fv,t dv. Convergence in L follows therefore by coupling the uniform bound on the regularity of the solution with the time decay of the Fourier metric [,6] fξ,t ĝξ,t ˆ d f, gt = sup ξ 0 ξ. This strategy is popular in the field of nonlinear diffusion equations cf. [5,3] and the references therein, where, after scaling to obtain confinement, convergence in relative entropy is used to deduce rates of convergence in more standard norms, like L,orinC k by interpolation. The results in [3], which are referred to fast diffusion equations, are linked to the present problem for one additional aspect, namely the fat tails of the asymptotic profile, which possesses only moments of low order. A related problem for a linear Fokker Planck equation with coefficient of diffusion growing with the distance from the origin is treated in [0]. There, convergence in Fourier weak norm coupled with regularity has been used to prove strong convergence to equilibrium in the physical space. Last, we recall that other weak norms can be used to control the convergence of kinetic equations towards equilibrium, among others the wellknown Wasserstein metric []. This metric has been successfully used to control convergence towards equilibrium in Fokker Planck models of granular media [3,]. 9 0

4 7 G. Furioli et al. / Ann. I. H. Poincaré A Going back to the results of [], their validity requires the assumption of weak inelasticity, which corresponds to fix the coefficient of restitution e sufficiently close to. A different technique which allows to control the growth of the frequencies of the Fourier transform of the solution has been recently applied in [8] to a one-dimensional dissipative model introduced by Ben-Avraham and coworkers []. The results in [8] are independent of the degree of dissipation. By adapting this technique to the three-dimensional case, we will end up with a result which is independent of the coefficient of restitution e. The starting point of this analysis is the results of the paper [], where the proof of uniform propagation of regularity makes use of the following property for the solution ftof the elastic Boltzmann equation sup ξ R ˆ fξ,t 0, R + uniformly in time. This property is proved by exploiting the pointwise convergence of the Fourier transform of the solution to the Maxwellian equilibrium ˆMξ = e ξ / and the decreasing property with respect to ξ ofthe Maxwellian itself. In the dissipative case, condition 3 cannot be satisfied by ftbut still holds for the solution to the scaled Boltzmann equation gt, provided the Fourier transform of the initial data satisfies μ + κ ξ ĝ0 ξ K, ξ R 3 for some positive constants κ, K and μ. Indeed, we prove that the solution gt satisfies an analogous estimate + κ ξ μ ĝξ,t K, ξ 5 uniformly in time, for possibly different κ, K and μ. Condition 5 is difficult to prove directly from the equation satisfied by the scaled density gt, due to the presence of the drift term in Eq.. The key point of our approach is to consider a semi-implicit discretization of Eq., where the drift term is absorbed in an integral term which is much easier to handle. The same trick has been successfully exploited in the one-dimensional case in [8] where it has been also pointed out that an equivalent approach could have been through the Duhamel formula for the solution gt see e.g. [6]. Our main results are summarized into the following statements. Theorem. Assume e [0,. Let gt be the weak solution of Eq., corresponding to the initial probability density g 0 with zero mean velocity, g 0 v dv = 3 and such that g 0 v dv<+. If in addition ĝ 0 ξ, ξ >R, + β ξ ν 6 for some R>0, ν>0and β>0, then there exist ρ>0, k>0, ν > 0 such that gt satisfies ĝξ,t, ξ ρ, t 0, + k ξ 7, ξ >ρ, t 0. + β ξ ν Theorem. Assume e [0,. Let gt be the weak solution of Eq., corresponding to the initial probability density g 0 with zero mean velocity, g 0 v dv = 3 and satisfying g 0 v dv <+. Let us suppose moreover g 0 H η for some η>0 and g 0 H ν for some ν>0. Then gt converges strongly in L with an exponential rate towards the stationary solution g, i.e., there exist positive constants C and γ explicitly computable such that gt g L Ce γt, t 0. 3 Thanks to the scaling invariance of the L -norm, Theorem allows to deduce also the strong convergence of the original non-scaled solution fτto the self-similar state fτ f τ e L Ce γ 8ε τ, τ 0,

5 G. Furioli et al. / Ann. I. H. Poincaré A where f v, τ = 3 3/ 3 / g v Tτ Tτ for Tτas in 5 and this independently of the initial temperature T 0 of f 0.. Preliminary results Following Bobylev [] it is convenient to rewrite Eq. for Maxwell molecules in the dissipative case in the Fourier variables: fξ,τ= τ ˆ f ˆ ξ +,τ f ˆ ξ,τ fξ,τ ˆ fξ,0 ˆ dσ, 8 πε where σ S ξ + = 3 e ξ + + e ξ σ, ξ = + e ξ ξ σ = ξ ξ +. 9 The existence and uniqueness of a solution of 8 for any initial data f 0 satisfying can be established through the application of Wild sums [6]. Theorem 3 Theorem of existence and uniqueness [,6]. We consider f 0 0 satisfying the normalization conditions and the following Cauchy problem: fξ,τ= τ ˆ f ˆ ξ +,τ f ˆ ξ,τ fξ,τ ˆ fξ,0 ˆ dσ, τ > 0, πε 0 σ S fξ,0 ˆ = fˆ 0 ξ. Then, there exists a unique nonnegative solution f C [0, +, L to Eq. 0. This solution preserves mass and momentum, while the energy decays at the exponential rate given by 5. Let gv,t be defined by 0. One can easily show that gv,t preserves the temperature, and moreover gv,tdv = 3, t 0. Moreover if the initial data g 0 has a diagonal pressure tensor which is unitary v i v j g 0 v dv = δ i,j, i,j =,, 3, then all the second moments are also preserved. If on the contrary the pressure tensor is not unitary, then its nonisotropic part v i v j gv,tdv for i j vanishes if it is initially vanishing cf. [], whereas the isotropic part vi gv,tdv is in general not preserved. evertheless, the condition g 0 v dv = 3 implies that the matrix [ v i v j g 0 v dv] i,j is real and symmetric. Consequently there exists a suitable orthonormal system in in which it is diagonal and therefore it remains diagonal for all t>0. Owing to this property, throughout this paper we will assume, without any additional assumption, that g 0 has diagonal pressure tensor, although not unitary. It is well known that in Maxwell models the time evolution of moments can be evaluated exactly. In particular, this can be done for the diagonal terms vi gv,tdv. Wehave

6 7 G. Furioli et al. / Ann. I. H. Poincaré A Proposition. Assume e [0,. Let gt be the weak solution of Eq., corresponding to the initial probability density g 0 with zero mean velocity, diagonal pressure tensor and satisfying g 0 v dv = 3. Then, we have R 3 v gv,tdv [ ] [ ] [ ] v gv,tdv = + C e +e e t + C 0 e +e e t, t 0, v3 gv,tdv 0 where C = v g 0v dv and C = v 3 g 0v dv. Proof. Recalling Eq., the expression 3 of the post-collisional variables, the conservations of the mass and the vanishing both of the momentum and of the non-isotropic terms v i v j gv,tdv along the solution, we obtain the following linear differential system: d dt R 3 v gv,tdv v gv,tdv = v3 gv,tdv 3 +e +e e 3 e +e 3 e 3 +e e +e 3 e +e 3 e +e 3 e +e 3 e 3 +e e R 3 v gv,tdv v gv,tdv. v3 gv,tdv [ ] Since the matrix of coefficients has 0 as simple eigenvalue and +e e as double eigenvalue, with eigenspaces [ ] [ ] and, 0 respectively, the general solution of the system is 0 R 3 v gv,tdv [ ] [ ] [ ] v gv,tdv = + C e +e e t + C 0 e +e e t, t 0, v3 gv,tdv 0 for C, C real constants to be determined by the initial conditions. In Fourier variables, the function gt satisfies the equation t ĝξ,t ξ ξ ĝξ,t = E ĝ ξ +,t ĝ ξ,t ĝξ,tĝξ,0 dσ, π σ S ĝξ,0 =ĝ 0 ξ. We will denote Q + ĝ,ĝξ, t = π σ S ĝ ξ +,t ĝ ξ,t dσ. Hence Eq. 3 can be written t ĝξ,t ξ ξ ĝξ,t = E Q + ĝ,ĝξ, t ĝξ,t. For s>0letp s define the set of probability densities satisfying s fvdv<+. 3 Consider on P s the distance fξ ĝξ ˆ d s f, g = sup ξ 0 ξ s. It is not difficult to show through a Taylor expansion that d s f, g is finite for any pair of probability densities f and g with equal moments v β fvdv for any multi-index β 3 of length smaller than or equal to [s] if s, itis enough to suppose equal moments up to the s -th order. For α 0, ] and initial data g, g P +α which

7 G. Furioli et al. / Ann. I. H. Poincaré A share the same moments up to the second order and have unitary pressure tensor as in, it is possible to estimate the d +α distance between the two solutions as follows. Theorem 5 Strict contraction of d +α []. Assume e [0,. Forα 0, ] there exists an explicit constant Cα,e > 0, Cα,e 0 as α 0, such that for any g t and g t solutions of corresponding to initial values g 0, g0 in P +α with unit mass, zero mean velocity and unitary pressure tensor, then d +α g t, g t d +α g 0,g 0 e Cα,et, t 0. In our framework, the constant Cα,e has the following expression: Cα,e = E Aα, e + α 5 where Aα, e = ξ + +α + ξ +α π ξ +α dσ. 6 S A detailed analysis of Cα,e can be found in []. It is worth noticing that we cannot deduce from the previous theorem the uniform boundedness in time neither of +α g v, t dv nor of +α g v, t dv. evertheless, if the initial data g 0 belongs to P, it is possible to prove that the solution keeps on satisfying the same property uniformly in time. Theorem 6 Uniform control of th moment [9]. If g 0 is a Borel probability density in P then the solution gt to with initial data g 0 belongs to P for all t 0 and sup gv,tdv<. t0 Thanks to the uniform boundedness of the fourth moment, it is possible to prove, via a fixed point argument, the existence of a universal stationary state in a suitable subspace of P. Moreover, the strict contraction of the d +α metric allows also to prove the stability of this stationary state in this subspace. Theorem 7 Stationary states [5,,6]. Let e [0, be fixed. Eq. has a unique stationary state g which is a probability density in { H = g 0, gvdv =, v i gvdv = 0, v i v j gvdv = δ i,j,i,j=,, 3, R } 3 +α gvdv<+, for α 0, ]. This stationary state is a radial function and belongs to P with g v dv M, where M depends only on e [0,. Moreover, for α 0, ], given any gt solution of issued from an initial datum g 0 P +α with unit mass, zero mean velocity and unitary pressure tensor, then d +α gt,g d+α g 0,g e Cα,et, t 0, where Cα,e is the constant 5. The hypothesis of unitary pressure tensor can in fact be removed. Under the only assumption that the initial datum belongs to P +α with g 0 v dv = 3 and same moments up to the first order as the stationary state, the d distance between the solution and the stationary state is proved to be exponentially decreasing.

8 76 G. Furioli et al. / Ann. I. H. Poincaré A Theorem 8 Stability in d without unitary tensor pressure [,6]. Let e [0, be fixed. For any α 0, ] and any initial datum g 0 P +α with unit mass, zero mean velocity and g 0 v dv = 3, there exist positive constants K and K depending on e,α,g 0 such that given gt the solution of 3 issued from g 0 and g the stationary state, then d gt,g K e K t, t 0. As a consequence, the uniqueness of the stationary state holds in a larger space where the second moments are not prescribed. Corollary 9 Uniqueness of stationary states [5,6]. The uniqueness of the stationary state g found in Theorem 7 holds true in { H = g 0, gvdv =, v i gvdv = 0, i=,, 3, gvdv = 3, R } 3 +α gvdv<+, for α 0, ]. As far as the smoothness of the stationary state is concerned, it is known that g H s, for all s 0. This is a consequence of the following result. Theorem 0 Smoothness of the stationary state [5], Theorem 5.3. For e [0,, the stationary state g satisfies the bounds e ξ ĝ ξ + ξ e ξ, ξ. 3. The iteration process The goal of this section is to build up a sequence of functions {g ξ, t} which approximates uniformly the solution ĝξ,t. In order to do this, for any fixed T>0weconsider first a semi-implicit discretization in time of Eq. 3 by partitioning the interval [0,T] into subintervals and we define thus the approximate solution at any time t = j T for j = 0,...,. Second, we define g ξ, t on the whole interval [0,T] by interpolation and last we show the convergence of the approximation to the solution. Other details and the missing proofs of this section can be found in [8]. We begin by introducing a semi-implicit discretization in time of Eq. 3 as follows. Let T>0and t = T for, >T.Let ˆφ j ξ, j = 0,...,, be the sequence: ˆφ 0 ξ =ĝ 0ξ, ˆφ j+ ξ ˆφ j ξ = ξ ξ ˆφ j+ t + E Q + ˆφ j, ˆφ j 7 ξ ˆφ j ξ, j = 0,...,. Proposition. Assume e [0,.Ifg 0 is a probability density with zero mean velocity satisfying g 0 v dv = 3, then there exists a unique sequence of bounded functions ˆφ j for j =,...,, satisfying 7. This sequence is defined as follows ˆφ 0 ξ =ĝ 0ξ, ˆφ j+ ξ = + EtQ+ ˆφ j t, ˆφ j ηξ + Et ˆφ j ηξ dη, j = 0,...,. 8 η t +

9 Since η ξ cos θ sin ψ,η ξ sin θ sin ψ,η ξ cos ψ = ηξ, G. Furioli et al. / Ann. I. H. Poincaré A Proof. Let us begin by proving that ˆφ is well defined. We proceed in the same way as in Proposition 7 in [8]. For ξ 0 we multiply Eq. 7 by t ξ t to obtain ξ t ˆφ t ξ + ξ t ξ ξ ξ ˆφ ξ = ξ t EtQ + ĝ 0, ĝ 0 ξ + Etĝ 0 ξ, t or, what is the same ξ t ˆφ ξ ξ = ξ t EtQ + ĝ 0, ĝ 0 ξ + Etĝ 0 ξ. t In spherical coordinates, ξ = ξ cos θ sin ψ, ξ sin θ sin ψ, ξ cos ψ, where θ [0, π and ψ [0,π]. Since ĝ 0 ξ is bounded, integrating over [ ξ, + we get ξ t ˆφ ξ = t + ξ EtQ+ ĝ 0, ĝ 0 s,θ,ψ+ Etĝ 0 s,θ,ψ s t ds. through the change of variables η = ξ s we finally obtain ˆφ ξ = t + EtQ+ ĝ 0, ĝ 0 ηξ + tĝ 0 ηξ dη. 9 η t + Since the initial density g 0 has unit mass, zero mean velocity and bounded temperature, then ĝ 0 belongs to C and there exists C>0such that ĝ 0 0 =, k ĝ 0 ξ C, k ĝ 0 0 = 0, k=,, Therefore the function ˆφ defined by continuity in ξ = 0as ˆφ 0 = is the unique bounded and C solution of 7. By an iteration argument the same conclusion holds for any ˆφ j. Hence, for j = 0,..., it holds ˆφ j+ ξ = t + EtQ+ ˆφ j, ˆφ j ηξ + Et ˆφ j ηξ dη η t +. Remark. We remark that Fubini s theorem implies that any function ˆφ j+ ξ is the Fourier transform of φ j+ v, where for j = 0,..., and v φ0 v = g 0v, φj+ v = t + Et η 3 Q + φ j,φ j v + Et η η 3 φ j v dη η. 3 η t + In the following proposition we gather the results on the moments of the approximation φj which will be used in the proof of Theorem. For the sake of clarity, the proof of the proposition will be postponed to Appendix A. Proposition 3. Assume e [0, and let g 0 be a probability density in P with zero mean velocity, satisfying v i v k g 0 v dv = 0, i k, g 0 v dv = 3. 3

10 78 G. Furioli et al. / Ann. I. H. Poincaré A Let φj be the approximation defined in 3. Then, there exists C>0such that for large enough, for j = 0,..., it holds φ j v 0, v i v k φ j v φ j v φ j φ j v dv =, v dv = 0, i k, v dv v dv v3 φ j v dv φj v dv C, = [ ] + C [ 0 v k φ j ] t 3 e e t where C = v g 0v dv and C = v 3 g 0v dv. v dv = 0, k=,, 3, 33 j + C [ 0 3 ] t 3 e j e, 35 t We define the approximate solution g ξ, t at any time t = j T as g ξ, j T = ˆφ j ξ for j = 0,...,.We extend afterwards the definition on the whole interval by interpolation. More precisely, let us define { ĝ0 g ξ, t = 0, ξ, t = αt ˆφ K ξ + αt ˆφ K ξ, 0 <tt, where for 0 <tt and K {,...,} we have K T <tk T. Thus there is a function 0 αt < such that t = αtk T + αtk T.Anyg ξ, t is continuous on [0,T] and for any t [0,T] it belongs to C. The result of convergence is therefore as follows. Proposition. There is a subsequence {g l ξ, t} l of {g ξ, t} which converges uniformly on any compact set of [0,T] to the solution ĝξ,t.. Control of tails of the Fourier transform of the solution In this section we prove Theorem. Thanks to the uniform convergence of a subsequence of the approximate solutions g ξ, t to the solution ĝξ,t and to the definition of g ξ, t, it is enough to prove bounds 7 for any ˆφ j ξ uniformly for large enough and j = 0,...,. We remark that the control of low frequencies follows directly from properties and from the boundedness of the fourth moment of the initial data. Lemma 5. Assume e [0,. Let gt be the weak solution of Eq., corresponding to the initial probability density g 0 P with zero mean velocity and g 0 v dv = 3. Let ˆφ j be the approximation defined in 8. There exist k>0 and ρ>0 such that for any fixed T>0 and any large enough we get ĝξ,t, + k ξ ξ ρ, t 0, 37 ˆφ j ξ, + k ξ ξ ρ, j = 0,...,. 38 Proof. Both estimates will be achieved through a MacLaurin expansion in which all terms will be bounded uniformly thanks to Proposition, Theorem 6 and Proposition 3. In what follows, constants may vary from one line to another. 36

11 G. Furioli et al. / Ann. I. H. Poincaré A Let us begin by proving 37. From the hypotheses on g 0 and Proposition, the MacLaurin expansion in Fourier variables reads ĝξ,t = 3 k= vk gv,tdv ξk + 0 D 3 ĝsξ,tξ,ξ,ξds, ξ,t 0. For all i, j, k =,, 3 and all t 0 the conservation of the mass and the uniform boundedness of the fourth moment Theorem 6 implies v i v j v k gv,tdv C<+. On the other hand, for all i, j, k =,, 3 and all t 0 3 ij k ĝξ,t C v i v j v k gv,tdv. Hence we have the uniform upper bound D 3 ĝsξ,tξ,ξ,ξds C ξ 3, ξ,t0. 0 The time evolution of the second moments of gt obtained in Proposition implies a strictly positive uniform lower bound on vk gv,tdv for k =,, 3 and t 0. Therefore, for any ξ R3 and t 0 we have the estimate ĝξ,t C ξ + D ξ 3, 39 with C and D suitably chosen. Inequality 39 implies that there exist ρ>0and k>0 such that ĝξ,t, ξ ρ, t 0. + kξ To prove 38, thanks to the uniform estimates collected in Proposition 3, we can proceed exactly in the same way. We are now in position to prove Theorem. Theorem. Assume e [0,. Let gt be the weak solution of Eq., corresponding to the initial probability density g 0 P with zero mean velocity and g 0 v dv = 3. If in addition ĝ 0 ξ, ξ >R, + β ξ ν 0 for some R>0, ν>0and β>0, then there exist ρ>0, k>0, ν > 0 such that gt satisfies ĝξ,t, ξ ρ, t 0, + k ξ, ξ >ρ, t 0. + β ξ ν Proof. The bound on the low frequencies ξ ρ has been established in Lemma 5. Moreover, in consequence of Proposition 3.3 in [7], we can suppose that condition 0 holds for any ξ >ρwith a possibly smaller exponent ν. We will now prove that, for any and j = 0,...,, ˆφ j ξ, ξ >ρ, + β ξ ν

12 730 G. Furioli et al. / Ann. I. H. Poincaré A for a positive constant ν small enough. By induction we have only to check the bound on ˆφ ξ = t + η= EtQ+ ĝ 0, ĝ 0 ηξ + Etĝ 0 ηξ dη η t +. Let ξ >ρand η>. We start to bound the term Q + ĝ 0, ĝ 0 ηξ = ξ + ĝ 0 ξ dσ, π S ĝ 0 where ξ = ηξ. Since η >, we have ξ =η ξ > ξ >ρ. We recall that for σ S ξ + = 3 e ξ + + e ξ σ, ξ = + e ξ ξ σ, so that ξ + + ξ = ξ. Therefore ξ + + ξ ξ, and so either ξ + ξ or ξ ξ. Moreover, for any σ S ξ + e e ξ ξ. 3 The surface S can be split into six non-overlapping domains S = A 0 A A A 3 A A 5, where A 0 = { σ S : ξ + >ρ, ξ >ρ }, A = { σ S : ξ + ρ, ξ } >ρ, { A = σ S : ξ ρ, ξ ξ, ξ + } ξ >, { A 3 = σ S : ξ ρ, ξ ξ >, ξ + ξ >, ξ + ρ { A = σ S : ξ ρ, ξ > ξ, ξ + > ξ, ξ + >ρ { A 5 = σ S : ξ ρ, ξ + ξ }. We use the aforementioned decomposition to estimate Q + ĝ 0, ĝ 0 ηξ 5 ĝ 0 ξ + ĝ 0 ξ dσ. π If σ A 0, inequality gives ĝ 0 ξ + ĝ 0 ξ Hence A 0 i=0 A i + β ξ + ν + β ξ ν }, }, + β ξ + + ξ ν + β ξ ν. ĝ0 ξ + ĝ 0 ξ dσ + β ξ ν A 0,

13 G. Furioli et al. / Ann. I. H. Poincaré A where A 0 denotes the Lebesgue measure of A 0. Let us choose now σ A. Thanks both to inequality 3 and to the lower bound ξ ξ which holds true since ξ ξ +, we obtain ĝ 0 ξ + ĝ 0 ξ + k ξ + + β ξ ν + k e ξ + β. ξ ν On the other hand, if ν > 0 is small enough + k e ξ + β. ξ ν + β ξ ν Indeed, since + x/ + x for any x 0 + β ξ A + β ξ ν + k e ξ ν + k e + k e ρ ξ + β ξ ν where the last term is smaller than for ν log + k e ρ. Finally ĝ 0 ξ + ĝ 0 ξ dσ A. 5 + β ξ ν Let us now fix σ A. We get ĝ0 ξ + ĝ 0 ξ dσ + β ξ + ν A + + k ξ + A { ξ + >ρ} A { ξ + ρ} ĝ0 ξ dσ Thanks to 3, ξ + >ρ e, so that for ν small enough we obtain + k ξ + + β ξ +. ν Using the previous bound into 6 we conclude A ĝ 0 ξ + ĝ 0 ξ dσ + β ξ + ν A ĝ 0 ξ dσ ĝ 0 ξ dσ. 6 + β ξ ν A + k ξ dσ. To evaluate the integral over A, we introduce a spherical reference frame centered at the collision center between the two particles with z-axis defined by ξ or ξ, which is just a multiple. We will denote ϑ [0,π] the longitude and ϕ [0, π the latitude of the point on the sphere corresponding to the vector σ. In this reference frame ξ + = 0 + e e ξ 3 e + e + ξ cos ϑ, 6 8 ξ + e = ξ cos ϑ. Consequently { ρ A = ϕ [0, π, ϑ [0,π]: cos ϑ min +e ξ, 6 3 e + e, 8 +e }.

14 73 G. Furioli et al. / Ann. I. H. Poincaré A Since in A ξ + > ξ +e and 6/[3 e + e] /[8 ] for all e>0, { ρ A = ϕ [0, π, ϑ [0,π]: cos ϑ min +e ξ, Let us suppose first ξ such that ξ ρ. In this case, we have + k ξ dσ = A If on the contrary ξ < ρ then A ϕ [0,π ρ cos ϑ +e ξ = + k ξ dσ = = 8 +e }. + k +e ξ cos ϑ sin ϑ dϑ dϕ π ρ dx +e ξ + kx = πρ log + kρ +e ξ kρ = log + kρ kρ A. ϕ [0,π π +e ξ 0 cos ϑ 8 +e ξ 0 + k +e ξ cos ϑ dx + kx = π +e ξ = π log + k 8 +e k ξ In both cases we get log + kt dσ sup A + k ξ kt A so that A t ρ ĝ 0 ξ + ĝ 0 ξ dσ + β ξ ν sup t ρ By choosing ν > 0 small enough log + kt + β sup. ξ ν kt + β ξ t ρ ν Indeed, + β ξ ν + β ξ as soon as ν log inf t ρ sup t ρ log + kt kt kt. log + kt ν sup t ρ log + k ξ k ξ log + k = k ξ log + kt A. kt log + kt kt, A. ξ sin ϑ dϑ dϕ

15 G. Furioli et al. / Ann. I. H. Poincaré A Finally A ĝ0 ξ + ĝ 0 ξ dσ + β ξ ν A. 7 Let us fix now σ A 3.Forν > 0 small enough it holds ĝ 0 ξ + ĝ 0 ξ + k ξ + + k ξ + k ξ + + ξ + k ξ + β ξ ν. Thus A 3 ĝ 0 ξ + ĝ 0 ξ dσ A β ξ ν For σ A and ν > 0 small enough ĝ 0 ξ + ĝ 0 ξ + β ξ + ν + k ξ + β ξ ν + k ξ Therefore A + β ξ ν. ĝ 0 ξ + ĝ 0 ξ dσ A. 9 + β ξ ν Last, let us consider σ A 5. It is worth noticing that { A 5 ξ ξ >, ξ + } ρ so that, for ν > 0 small enough ĝ 0 ξ + ĝ 0 ξ + k ξ + + k ξ + k ξ This implies A 5 + β ξ ν. ĝ 0 ξ + ĝ 0 ξ dσ A β ξ ν Grouping inequalities, 5, 7, 8, 9 and 50 we conclude that it exists a suitably small positive constant ν depending on ρ and k fixed once and for all in the first part of this proof such that, for ξ >ρand η> Q+ ĝ 0, ĝ 0 ηξ S =. π + β ξ ν + β ξ ν Coming back to ˆφ ξ for large enough and ξ >ρwe obtain ˆφ ξ t + η= Et Remark 6. ote that the bounds in imply ĝξ,t C + κ ξ μ, ξ R3,t0. + Et + β ξ ν + β ξ ν dη η t + = + β ξ ν.

16 73 G. Furioli et al. / Ann. I. H. Poincaré A Propagation of regularity and strong convergence The aim of this section is to prove Theorem about the strong L -convergence of the solution gt of Eq. to the stationary state g. Theorem. Assume e [0,. Let gt be the weak solution of Eq., corresponding to the initial probability density g 0 P with zero mean velocity and g 0 v dv = 3. Let us suppose moreover g 0 H η for some η>0 and g 0 H ν for some ν>0. Then gt converges strongly in L with an exponential rate towards the stationary solution g, i.e., there exist positive constants C and γ explicitly computable such that gt g L Ce γt, t 0. Let us begin by the following lemma cf. [0], which makes a link between the condition g 0 H ν and the decay of the Fourier transform. For the proof we refer to [8]. Lemma 7. Let g 0 be a probability density such that g 0 H ν for some ν>0. Then there exist positive constants C and β such that ĝ 0 satisfies ĝ 0 ξ C + β ξ ν, ξ R3. In order to prove Theorem, we need to pass from the weak convergence in the Fourier distance d of the solution gt to the stationary state g Theorem 7 to the more natural L -convergence in the physical space. To this extent, we make use of the propagation of regularity established in Theorem. The additional ingredient is the uniform boundedness of the solution in Sobolev norms. The proof of this statement follows along the lines of the analogous proof in [], Lemma 3. and Theorem 3.6. In [], however, the decay of the Fourier transform of the solution was proved only for small inelasticity e, making use of a precise control of the growth of the Fisher information Theorem.. Theorem 8. Assume e [0,. Let gt be the weak solution of Eq., corresponding to the initial probability density g 0 P with zero mean velocity and g 0 v dv = 3. Moreover, let g 0 H η for some η>0 and g 0 H ν for some ν>0. Then gt is uniformly bounded in H η. Proof of Theorem. The proof of Theorem is completed using the following interpolation bounds: there exists a positive constant C such that h L C h 3 7 h L 7 ; L for any s 0 and any η>0, there exists a positive constant C such that h H s C sup ξ 0 ĥξ ξ η s+7+η h η s+7+η H. s+η To prove the last bound, consider that, for any given constant R>0 h = H ĥξ ξ s ĥξ dξ = s ξ ξ s+ dξ + ĥξ ξ s+η ξ η dξ ξ R ξ >R ĥξ C sup ξ ξ R s+7 + Rη h. H s+η Optimizing over R gives the result.

17 G. Furioli et al. / Ann. I. H. Poincaré A Let h = gt g. Keeping in mind the regularity of the stationary state Theorem 0, for s = 0 we get gt g L C gt L + g L 3 7 Appendix A η d gt,g 7+η gt η H η + g H η 7+η 7. Proof of Proposition 3. The estimates on the first order moments 33 are easily obtained by a recursive procedure based on expression 3. A similar procedure applies to show that the non-isotropic second moments 3 vanish. Let us now consider the diagonal second moments 35. Using the same procedure of Proposition we arrive at the following first order linear difference system where v φ j+ dv v φ j+ dv = v3 φ j+ dv A = 3 e e +e 3 e +e 3 e +e 3 e 3 e e +e 3 e t v φ j dv Id + ta v φ j dv v3 φ j dv +e 3 e +e 3 e 3 e e. [ ] The matrix A has as simple eigenvalue, with eigenspace and 3 e e as double eigenvalue with eigenspace [ ] [ ], 0. Therefore the matrix t Id + ta has as simple eigenvalue and 3 e t t e as double 0 eigenvalue, with the same eigenspaces. Since for large enough depending on e [0, wehave0< t t 3 e e <, we obtain v φ j v dv [ ] [ ] v t 3 e [ ] j φ e t 3 e j e j v dv = + C + C 0, 0 t t v3 φ j v dv with C, C real constants determined by the initial conditions. By virtue of 35, for all and for all j = 0,..., φj v dv = 3. Finally, let us take into account the fourth moment 36. Making use again of 3 we find φj+ v dv = t We denote m,j = + Et = t t Et η 3 Q + φ j,φj φj v dv. Q + φ j,φj v dv + Et η v dv + Et η 3 φ j v dη dv η η t + φj v dv.

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