Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibriaThe Patterson-Sullivan embedding and minimal volume entropy for outer space

CONTRACTIVE METRICS FOR A BOLTZMANN EQUATION FOR GRANULAR GASES: DIFFUSIVE EQUILIBRIA. M. BISI, J. A. CARRILLO AND G. TOSCANI Abstract. We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t→ ∞ to this diffusive equilibrium in this distance metrizing the weak convergence of measures. Then, we prove a uniform bound in time on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. These results are then combined, using interpolation inequalities, to obtain exponential convergence to the diffusive equilibrium in the strong L1-norm, as well as various Sobolev norms. 1. Introduction This paper concerns the long-time asymptotics of the the homogeneous Boltzmann equation for inelastic interactions in the pseudo-Maxwellian approximation introduced in [4]. Without an energy source into the system, the inelasticity of the collisions will homogenize the velocity of particles and thus, the only possible steady state is the Delta Dirac distribution at the mean velocity of the particles [4]. Therefore, we assume that our particles are under the influence of an external source of energy, a thermostat, that keeps the motion of particles. A stochastic thermostat was indeed proposed in different granular gases models as a candidate for keeping the temperature of particles bounded away from zero [1, 2, 22]. From a probabilistic point of view, a stochastic thermostat means that particles follow Brownian paths between collisions. At the kinetic level the thermostat is represented by a linear diffusion term in velocity [11], that may be temperature dependent [20]. Hence, if f(v, t)≥0 denotes the density for the velocity space distribution of the molecules of the granular gas at time t, the time evolution of fobeys the 1 2 M. BISI, J. A. CARRILLO AND G. TOSCANI equation ∂f ∂t =Bpθ(t)Q(f, f) + F θp(t)4vfwith 0 ≤p < 3 2,(1.1) where Q(f, f), the Boltzmann collision operator in the pseudo-Maxwellian approximation, models the effects of inelastic binary collisions between particles. We keep the positive constants Band F, measuring the importance ratio of inelastic collisions and stochastic thermostat to keep track of their influences in the convergence rates. Instead of writing explicitly the collision operator [4], it is much easier (and less painful) to work in the corresponding weak form of the Q(f, f). Thus, we will define Q(f, f) by its weak form given by hϕ, Q(f, f)i=1 4πZR3ZR3ZS2 f(v)f(w)hϕ(v0)−ϕ(v)idn dv dw (1.2) acting on functions ϕ∈C(R3) where v0=1 2(v+w) + 1−e 4(v−w) + 1 + e 4|v−w|n(1.3) is the postcollisional velocity and 0 < e < 1 is the constant restitution coefficient. The unit vector nin (1.3) parameterizes the set of all kinematically possible (i.e. those conserving mass and momentum) post-collisional velocities. Without loss of generality, in what follows we will assume that the initial datum f0is a probability density. By mass conservation, f(v, t) will remain a probability density at any subsequent time. The existence of steady states for equation (1.1) has been addressed by fixed point arguments in [14] and by moment methods in [5]. Uniqueness of stationary solutions was pointed out as an open problem in [14] and later addressed indirectly in [5] as a consequence of the uniqueness of the moment expansion for solutions of the initial value problem to (1.1). Recently, the existence of homogenous cooling states (HCS) (self-similar solutions) without the diffusion term was established in [6] and subsequently, proved in [7] that they give the intermediate asymptotic of the solutions, that is, the difference between any solution and the HCS measured in a suitable way vanishes as t→ ∞. Let us finally mention that the mathematical properties of the IVP for the more realistic hard-spheres model with diffusion have been studied recently in [16] and its tail behavior well described both in the stationary and the time-dependent case [8, 16, 17]. However, the long-time asymptotic in this case are not well understood since even the uniqueness of steady solutions for a given mean velocity remains an open problem. In the rest of the paper, instead of working on the weak form (1.2), we will equivalently use the Fourier transformed equation corresponding DIFFUSIVE EQUILIBRIA 3 to (1.1) [3, 4], that reads ∂ˆ f ∂t =Bpθ(t)1 4πZS2½ˆ f(t, k−)ˆ f(t, k+)−ˆ f(t, 0) ˆ f(t, k)¾dn −F θp(t)|k|2ˆ f (1.4) with k−=1 + e 4(k− |k|n), k+=3−e 4k+1 + e 4|k|n . (1.5) Our convention with the Fourier transform is that ˆ f(k, t) = ZR3 f(v, t)e−iv·kdv. This weak form of equation (1.1) is suitable to show both the existence and uniqueness of a steady state and its regularity. A crucial role in our analysis is played by the weak norm convergence, which is obtained by adapting a method first used in [15] to control the exponential convergence for Maxwellian molecules in certain weak norms. The global asymptotical stability of the unique steady state can be recovered in terms of the contractivity properties of these distances in Fourier space. From their introduction on, these distances were subsequently used for the classical Boltzmann equation for Maxwellian molecules in [9, 21], for the Fokker-Planck equation in [12] and for the Kac model for inelastic interactions recently in [19]. Contractivity of equivalent metrics for solutions of a general family of PDE models, including simplified models of granular media, has recently been shown in [10, 18, 13] to produce results which are reminiscent of the main theorem in this work. Let Ps(R3) be the set of probability measures with bounded s-moment. The Fourier-based metrics ds, for any s > 0, are defined as ds(ˆ f, ˆg) = sup k∈R3 |ˆ f(k)−ˆg(k)| |k|s(1.6) for any pair of probability measures in Ps(R3). This distance is well-defined and finite by simple Taylor expansion for any pair of probability measures with equal moments up to order [s], where [s] denotes the integer part of s. Moreover, in case s≥1 be an integer, it suffices equality of moments up to order s−1 for being dsfinite. In fact, d2topology is equivalent to the weak-star topology for measures plus boundedness of second moments [21], and can be related to the well-known Wasserstein distance between probability measures. The main results of this work can be summarized as follows: 4 M. BISI, J. A. CARRILLO AND G. TOSCANI Theorem 1.1. Equation (1.1) has a unique steady state f∞in the set of probability measures with a given mean velocity u. Moreover, f∞is smooth and fast decaying at ∞. Furthermore, given any probability measure solution f(t, v)of (1.1) with mean velocity uand finite temperature, d2(ˆ f(t),ˆ f∞)tends exponentially to 0. The rate λof exponential convergence is explicitly computable. The key idea to prove this theorem is based on the observation that the d2distance is a contraction for the flow map of (1.1) restricted to initial data with equal temperature. This property will enable us to show the existence and uniqueness of the steady states by simple arguments invoking Banach fixed point theorem. The first step in our analysis will be the study of the time evolution of moments. We recover briefly the exact general moment equation of second order, and prove that the isotropic moments of any order satisfy a differential inequality which implies a uniform bound in time. These arguments will be described in Section 2. Section 3 deals with the existence of moments and regularity of the steady state. These properties are addressed directly on the integral equation in Fourier representation verified by the steady state itself. Section 4 is devoted to prove the exponential convergence towards steady state for any solution of (1.1). This is achieved by a careful estimate of the convergence of the temperature to the asymptotic steady value and using a variation of the argument proving the contractivity of the d2distance. Section 5 deals with the problem of obtaining strong convergence towards the steady state. This goal will be reached by establishing a propagation of smoothness result for (1.1). To state the result concisely, we introduce the Sobolev space norms k·kHr(R3), r≥0 by kfkHr(R3)=ZR3 |k|2r¯¯¯ ˆ f(k)¯¯¯ 2dk, then we prove Theorem 1.2. Let f0be any initial datum for equation (1.1) with kf0kHr(R3)finite. Then, any probability density solution f(t, v)of (1.1) is bounded in Hr, and there is a universal constant Crso that, for all t > 0, kf(t)kHr(R3)≤max ©kf0kHr(R3), Crª. The results of Theorems 1.1 and 1.2 will then be combined, using interpolation inequalities [9], to obtain exponential convergence to f∞in the strong L1-norm, as well as in various Sobolev norms. DIFFUSIVE EQUILIBRIA 5 Theorem 1.3. Let f0be any initial datum for equation (1.1). Let ² > 0be given. Then, there is a number rdepending only on ²so that whenever ZR3 |v|2rf0(v)dv +ZR3 |k|2r¯¯¯ ˆ f0(k)¯¯¯ 2dk < ∞ then it holds that kf(·, t)−f∞kL1(R3)≤C²exp {−(1 −²)λt}. Here, λis the rate of exponential convergence in d2distance, and C²is explicitly computable in terms of the integrals specified above. Moreover, increasing r, we obtain the same result if the L1-norm is replaced by any Hm-norm. 2. Evolution of moments Let us denote by ρ(t), u(t), θ(t) and pij(t) the density, mean velocity, temperature and pressure tensor components of the distribution function f. Since mass and mean velocity are preserved clearly by the collision operator (1.2) and we deal with the homogeneous problem, without loss of generality we assume in the rest of the paper that ρ(t) = 1 and u(t) = 0 for all t≥0. Following [5], we compute the general moment equations of second order. Lemma 2.1. [Evolution of moments [11, 5]]Let ˆ fbe a solution to (1.4) with unit mass, zero mean velocity and initial second order moments bounded, then ˆ fhas finite second order moments for any t > 0, dθ dt =−1−e2 4Bθ 3 2+ 2F θp,(2.1) and ∂ ∂t ZR3 f(v)vivjdv =−(1 + e)(3 −e) 8Bqθf(t)ZR3 f(v)vivjdv +δij 2µ1 + e 4¶2 B³θf(t)´3 2+δij 2F θp f(t) (2.2) for any t≥0. As a consequence, we have the following asymptotically stable steady value of the temperature θ∞=µ8F B(1 −e2)¶2 3−2p .(2.3) 6 M. BISI, J. A. CARRILLO AND G. TOSCANI Proof.- By multiplying (1.1) by vivjand integrating we get ∂ ∂t ZR3 f(v)vivjdv =B 4πqθf(t)ZR3ZR3ZS2 f(v)f(w) hv0 iv0 j−vivjidn dv dw +F θp f(t)ZR3 vivj4vf dv. Since v0=3−e 4v+1 + e 4w+1 + e 4|v−w|n we have ∂ ∂t ZR3 f(v)vivjdv =B 4πqθf(t)ZR3ZR3ZS2 f(v)f(w)("µ3−e 4¶2 −1#vivj +µ1 + e 4¶2³wiwj+|v−w|2ninj+|v−w|(winj+wjni)´ + µ(3 −e)(1 + e) 16 ¶³viwj+wivj+|v−w|(vinj+vjni) ´)dn dv dw +δij 2F θp f(t). Evolution equations (2.1) and (2.2) are then obtained imposing zero mean velocity and unit mass, while boundedness of second moments is an easy consequence of (2.1). Remark 2.2. [Temperature for the case p= 1] If p= 1, the evolution equation (2.1) for the temperature can be solved in analytic way. In fact, if we put z=θe−2F t, we have dz dt =−1−e2 4BeF t z3 2, from which with trivial calculations it follows: z(t) = ·z−1 2 0+1−e2 8 B F¡eF t −1¢¸−2 DIFFUSIVE EQUILIBRIA 7 namely the analytic expression of the temperature is θ(t) = "eF t θ−1 2 0+1−e2 8 B F(eF t −1)#2 .(2.4) For future reference we include here the computation: Zt 0pθ(τ)dτ =8 (1 −e2)Blog "1 + 1−e2 8θ−1 2 0 B F¡eF t −1¢#.(2.5) The exact evolution equations for moments of order higher than two has been done in [5] only for the third order. Evolution of higher order moments has been studied in [5] only in the relatively simple case of isotropic solutions. We obtain here an explicit inequality for the time evolution of isotropic moments of any order for general solutions, which leads to prove a uniform bound in time of these moments, in terms of the moments of the initial value. To simplify notations, in what follows we denote M2r(t) = ZR3 f(v, t)|v|2rdv, for any r∈N. Moreover, we will take advantage of the weak form of the collision integral hϕ, Q(f, f)i=1 8πZR3ZR3ZS2 f(v)f(w) hϕ(v0) + ϕ(w0)−ϕ(v)−ϕ(w)idn dv dw (2.6) acting on functions ϕ∈C(R3). Here (v0, w0) represent the post-collisional velocities, v0is given by (1.3), and w0=1 2(v+w)−1−e 4(v−w)−1 + e 4|v−w|n. (2.7) The weak form (2.6) has been considered in [4], and is equivalent to (1.2). We prove Lemma 2.3. [Uniform in time moment estimates] Let f(t, v)be the solution to equation (1.1), where the initial distribution f0(v)is such that ZR3 f0(v)|v|2rdv =cr<+∞(2.8) 8 M. BISI, J. A. CARRILLO AND G. TOSCANI for some r≥2. Then, M2r(t)satisfies the following differential inequality d dtM2r(t)≤ −BrM2(t) 3"1−e2r 4¡M2r(t) + M2(r−1)(t)M2(t)¢ −1 2 r−1 X m=1 µr m¶M2(r−m)(t)M2m(t)# +F M2(t)p(2r+ 4r2)M2(r−1)(t). (2.9) Consequently, M2r(t)is uniformly bounded in time. Proof.- Elementary computations show that 0≥(v0)2+ (w0)2−(v2+w2) = −1−e2 4£(v−w)2− |v−w|(v−w)·n¤, (2.10) and (v0)2+ (w0)2≥e2¡v2+w2¢.(2.11) Inequality (2.11) follows from (2.10). In fact, since 0≤£(v−w)2− |v−w|(v−w)·n¤≤2(v−w)2, we obtain (v0)2+ (w0)2≥(v2+w2)−1−e2 2(v−w)2=1 2(v+w)2+e2 2(v−w)2 ≥e2¡v2+w2¢. DIFFUSIVE EQUILIBRIA 9 Choosing ϕ(v) = |v|2r,r≥2, we obtain h|v|2r, Q(f, f)i= =1 8πZR3ZR3ZS2 f(v)f(w)h|v0|2r+|w0|2r− |v|2r+|w|2ridn dv dw =1 8πZR3ZR3ZS2 f(v)f(w)h(|v0|2+|w0|2)r−(|v|2+|w|2)ridn dv dw + r−1 X m=1µr m¶1 8πZR3ZR3ZS2 f(v)f(w)h|v|2(r−m)|w|2m − |v0|2(r−m)|w0|2midn dv dw ≤1 8πZR3ZR3ZS2 f(v)f(w)h(|v0|2+|w0|2)r−(|v|2+|w|2)ridn dv dw +1 2 r−1 X m=1 µr m¶M2(r−m)M2m. (2.12) Taking into account (2.11), we obtain 1 8πZS2 h(|v0|2+|w0|2)r−(|v|2+|w|2)ridn =1 8πZS2h|v0|2+|w0|2−(|v|2+|w|2)ir−1 X m=0 h|v0|2+|w0|2ir−m−1h|v|2+|w|2im dn ≤1 8πZS2h|v0|2+|w0|2−(|v|2+|w|2)ir−1 X m=0 e2(r−m−1)h|v|2+|w|2ir−1dn =−1−e2r 4h|v|2+|w|2ir−11 8πZS2£(v−w)2− |v−w|(v−w)·n¤dn =−1−e2r 8h|v|2+|w|2ir−1(v−w)2. (2.13) 16 M. BISI, J. A. CARRILLO AND G. TOSCANI Hence, the left-hand side is real, positive and finite for any R > 0. On the other hand, we may directly estimate this left-hand side. ZBRZS2 ˆ f∞(k−)ˆ f∞(k+)ˆ fc ∞(k)dn dk ≤ZBRZS2 |ˆ f∞(k−)|| ˆ f∞(k+)|| ˆ fc ∞(k)|dn dk ≤ZBRZS2 |ˆ f∞(k+)|| ˆ fc ∞(k)|dn dk ≤·ZBRZS2 |ˆ f∞(k+)|2dn dk¸1 2·4πZBR |ˆ f∞(k)|2dk¸1 2 . Now, we can change variables from kto k+. The equality ZBRZS2 |ˆ f∞(k+)|2dn dk =ZBRZS2 |ˆ f∞(k+)|2dk dk+ dn dk+, thanks to the formula dk+ dk =µ3−e 4¶2µ3−e 4+1 + e 4 k·n |k|¶≥µ3−e 4¶2µ1−e 2¶, implies ZBRZS2 |ˆ f∞(k+)|2dn dk ≤32 (3 −e)2(1 −e)4πZBR |ˆ f∞(k)|2dk. Collecting previous estimates, we finally obtain ZBRµ1+ 1−e2 8θ∞|k|2¶|ˆ f∞(k)|2dk≤µ32 (3 −e)2(1 −e)¶1 2ZBR |ˆ f∞(k)|2dk, (3.12) for any R > 0, that we may write as ZBR |ˆ f∞(k)|2dk ≤ZBR (A1−A2|k|2)|ˆ f∞(k)|2dk, with A1=µ32 (3 −e)2(1 −e)¶1 2 and A2=1−e2 8θ∞. Since the function (A1−A2|k|2)|ˆ f∞(k)|2is non-negative only in the ball of radius Ro= (A1/A2)1/2, we have ZBR |ˆ f∞(k)|2dk ≤ZBRo (A1−A2|k|2)|ˆ f∞(k)|2dk ≤A1ZBRo |ˆ f∞(k)|2dk . DIFFUSIVE EQUILIBRIA 17 Letting R→ ∞ proves that f∞∈L2(R3). Let us consider (3.12) again. It implies 1−e2 8θ∞ZBR |k|2|ˆ f∞(k)|2dk ≤µ32 (3 −e)2(1 −e)¶1 2ZBR |ˆ f∞(k)|2dk, for any R > 0, and thus f∞∈H1(R3). Higher order regularity follows by a recursive argument. Assuming f∞∈ Hm−1(R3), m≥2, let us prove that f∞∈Hm(R3). To this aim, multiply (3.11) by |k|2(m−1) ˆ fc ∞and integrate on BR. One gets 4π1−e2 8θ∞ZBR |k|2m|ˆ f∞(k)|2dk≤ ≤RBRRS2|k|2(m−1) ¯¯¯ ˆ f∞(k−)ˆ f∞(k+)ˆ fc ∞(k)¯¯¯dn dk. Proceeding as before and taking into account that |k+| ≥ 1−e 2|k|, one shows that there exists a constant C(m, e) such that ZBRZS2 |k|2(m−1) ¯¯¯ ˆ f∞(k−)ˆ f∞(k+)ˆ fc ∞(k)¯¯¯dn dk ≤C(m, e)ZBR |k|2(m−1)|ˆ f∞(k)|2dk, (3.13) for any R > 0. By induction, the right-hand side is uniformly bounded in R > 0. Collecting previous estimates, one proves as before that there exists a constant C(m, e, θ∞) such that ZBR |k|2m|ˆ f∞(k)|2dk ≤C(m, e, θ∞)ZBR |k|2(m−1)|ˆ f∞(k)|2dk, for any R > 0, proving the assertion. 4. Exponential decay of the d2distance. General case In the previous section, we proved exponential decay of the d2distance of any two solutions corresponding to initial data with the same temperature. Since the temperature θ∞of the steady state is a constant which does not depend on the temperature of the initial datum, this result only shows that there is exponential decay towards the steady state for initial values that have the same temperature of the steady state. Thus, Proposition 3.1 does not answer to the natural question of the decay towards the steady state of any solution, whatever the initial temperature could be. Here, we show 18 M. BISI, J. A. CARRILLO AND G. TOSCANI that this question can be answered positively. Indeed, the result of Proposition 3.1 can be extended to estimate explicitly the rate of convergence towards the unique equilibrium f∞in terms of d2for general initial data without the restriction on the second moment. First, we reckon the time decay of the ratio θ(t)/θ∞. Lemma 4.1. [Equilibration of temperature] Let z(t)denote the ratio θ(t)/θ∞.Then, for any constant γ > 0it holds zγ(t)−1≤c(γ) exp ½−µ3 2−p¶2F θp−1 ∞t¾,(4.1) if z0>1, while 1−zγ(t)≤c(γ) exp ½−µ3 2−p¶2F θp−1 ∞z 1 2 0t¾,(4.2) if z0<1. The constant c=c(γ)is given by c(γ) = 2γ 3−2pmax n1, zγ+p−3 2 0o¯¯¯z 3 2−p 0−1¯¯¯.(4.3) Proof.- From equation (2.1) it follows d dt µθ θ∞¶=−1−e2 4B θ 1 2 ∞µθ θ∞¶3 2 + 2F θp−1 ∞µθ θ∞¶p = 2F θp−1 ∞"−µθ θ∞¶3 2 +µθ θ∞¶p#. Hence z(t) satisfies the equation dz dt =−2F θp−1 ∞zphz3 2−p−1i,(4.4) from which we obtain easily d dt hz3 2−p−1i=−µ3 2−p¶2F θp−1 ∞z1 2hz3 2−p−1i.(4.5) Case z0>1. Since 3 2−p > 0, equation (4.4) shows that if z0>1, z(t) is non–increasing and moreover z(t)≥1 for all times. Consequently (4.5) implies the inequality d dt hz3 2−p−1i≤ − µ3 2−p¶2F θp−1 ∞hz3 2−p−1i, DIFFUSIVE EQUILIBRIA 19 which after integration gives z3 2−p−1≤³z 3 2−p 0−1´exp ½−µ3 2−p¶2F θp−1 ∞t¾. Hence z(t)≤½1 + ³z 3 2−p 0−1´exp ½−µ3 2−p¶2F θp−1 ∞t¾¾ 2 3−2p ,(4.6) and, for any power γ > 0 zγ(t)−1≤½1 + ³z 3 2−p 0−1´exp ½−µ3 2−p¶2F θp−1 ∞t¾¾ 2γ 3−2p −1. The right hand side of this last inequality is a function of the form y(x) = (1 + ax)q−1 where q=2γ 3−2p, a =z 3 2−p 0−1, x = exp ½−µ3 2−p¶2F θp−1 ∞t¾. (4.7) By Taylor formula y(x) = q(1 + a¯x)q−1ax; since in our case 0 ≤¯x≤1 it holds (1 + a¯x)q−1≤1 if q−1<0, (1 + a¯x)q−1≤(1 + a)q−1if q−1>0. Thus y(x)≤qmax n1,(1 + a)q−1oa x. (4.8) By inserting expressions (4.7) into (4.8), we find inequality (4.1) which shows the exponential decay of zγ(t)−1 for all values of γ > 0. Case z0<1. The proof is very similar to the previous one. If z0<1, equation (4.4) implies that z(t) is non–decreasing, so that z(t)≥z0for all times. Thus, equation (4.5) implies d dt h1−z3 2−pi≤ − µ3 2−p¶2F θp−1 ∞z 1 2 0h1−z3 2−pi, from which it follows 1−z3 2−p≤³1−z 3 2−p 0´exp ½−µ3 2−p¶2F θp−1 ∞z 1 2 0t¾. 20 M. BISI, J. A. CARRILLO AND G. TOSCANI Hence z(t)≥½1−³1−z 3 2−p 0´exp ½−µ3 2−p¶2F θp−1 ∞z 1 2 0t¾¾ 2 3−2p ,(4.9) from which it follows 1−zγ(t)≤1−½1−³1−z 3 2−p 0´exp ½−µ3 2−p¶2F θp−1 ∞z 1 2 0t¾¾ 2γ 3−2p . The function on the right hand side is of the form y(x)=1−(1 −ax)q, with q=2γ 3−2p, a = 1 −z 3 2−p 0, x = exp ½−µ3 2−p¶2F θp−1 ∞z 1 2 0t¾. (4.10) Taylor formula gives y(x) = q(1 −a¯x)q−1ax. Since 0 ≤¯x≤1, (1 −a¯x)q−1≤(1 −a)q−1if q−1<0, (1 −a¯x)q−1≤1 if q−1>0. Finally y(x)≤qmax n1,(1 −a)q−1oax, (4.11) and by inserting expressions (4.10) into (4.11) we find inequality (4.2). We remark that in this second case the decay is slower, and it depends on z0<1. Theorem 4.2. [Exponential decay of the d2distance] Any solution f(t, v)of (1.1) corresponding to an initial density with unit mass, zero mean velocity and finite initial temperature, converges exponentially towards the steady state f∞(v)in d2distance. More precisely, there exist constants C1, C2, C3>0such that d2(ˆ f(t),ˆ f∞)≤d2(ˆ f0,ˆ f∞) e−(1−γ0)C1t+ C2e−C3t,(4.12) for all t≥0, with γ0= (e2+ 3)/4<1. Proof.- We proceed exactly as in the proof of Proposition 3.1. Therefore we will only sketch the proof, and put the emphasis on the differences. In agreement with the notations introduced in Lemma 4.1, the variable θ(t) will be replaced by θ∞z(t). Since the temperatures of the two solutions are different, with respect to the computations of Proposition 3.1, we have DIFFUSIVE EQUILIBRIA 21 an additional term that we may write using the equation (3.10) satisfied by f∞. We have ∂ ∂t ˆ f(k)−ˆ f∞(k) |k|2=B 4πθ 1 2 ∞z1 2(t)ZS2"ˆ f(k−)ˆ f(k+)−ˆ f∞(k−)ˆ f∞(k+) |k|2 −ˆ f(0) ˆ f(k)−ˆ f∞(0) ˆ f∞(k) |k|2#dn −F θp ∞zp(t)|k|2ˆ f(k)−ˆ f∞(k) |k|2 +F θp ∞³z1 2(t)−zp(t)´ˆ f∞(k). (4.13) Equation (3.6) becomes in our case ¯¯¯¯ ∂th(t, k) + µB θ 1 2 ∞z1 2(t) + F θp ∞zp(t)|k|2¶h(t, k)¯¯¯¯ ≤γ0B θ 1 2 ∞z1 2(t)kh(t, ·)k∞+ϕ(t) (4.14) where h(t, k) = ˆ f(k)−ˆ f∞(k) |k|2, and ϕ(t) = F θp ∞¯¯¯z1 2(t)−zp(t)¯¯¯.(4.15) Proceeding as in Proposition 3.1, we obtain exp µB θ 1 2 ∞Zt 0 z1 2(s)ds¶kh(t, ·)k∞≤ kh(0,·)k∞+ Φ(t) +γ0B θ 1 2 ∞Zt 0 z1 2(τ)kh(τ, ·)k∞exp µB θ 1 2 ∞Zτ 0 z1 2(s)ds¶dτ, where Φ(t) = Zt 0 ϕ(τ) exp µB θ 1 2 ∞Zτ 0 z1 2(s)ds¶dτ. 22 M. BISI, J. A. CARRILLO AND G. TOSCANI By the generalized Gronwall lemma [21], denoting by w(t) the same quantity as in Proposition 3.1, we finally conclude that w(t)≤w(0) exp µγ0B θ 1 2 ∞Zt 0 z1 2(τ)dτ¶ +Zt 0 exp µγ0B θ 1 2 ∞Zt τ z1 2(s)ds¶ϕ(τ) exp µB θ 1 2 ∞Zτ 0 z1 2(s)ds¶dτ . (4.16) Hence kh(t, ·)k∞≤ kh(0,·)k∞exp µ−(1 −γ0)B θ 1 2 ∞Zt 0 z1 2(s)ds¶+ Ψ(t) (4.17) where Ψ(t) = exp µ−(1 −γ0)B θ 1 2 ∞Zt 0 z1 2(s)ds¶Zt 0 ϕ(τ) ×exp µ(1 −γ0)B θ 1 2 ∞Zτ 0 z1 2(s)ds¶dτ. (4.18) To finish the proof, it suffices to show that Ψ(t) vanishes for t→ ∞. First of all, notice that definition (4.15) implies that if p=1 2the function ϕ(τ) vanishes (thus no further proof is needed), while for p6=1 2 ϕ(τ)≤F θp ∞³¯¯¯z1 2(τ)−1¯¯¯+¯¯¯zp(τ)−1¯¯¯´. Therefore ϕ(τ) can be estimated using inequalities (4.1)–(4.2). DIFFUSIVE EQUILIBRIA 23 Case z0>1. The function Ψ(t) can be bounded in the following way: Ψ(t) = exp µ−(1 −γ0)B θ 1 2 ∞Zt 0hz1 2(s)−1ids −(1 −γ0)B θ 1 2 ∞t¶ ×Zt 0 ϕ(τ) exp µ(1 −γ0)B θ 1 2 ∞Zτ 0hz1 2(s)−1ids + (1 −γ0)B θ 1 2 ∞τ¶dτ ≤exp ³−(1 −γ0)B θ 1 2 ∞t´Zt 0 ϕ(τ) exp ³(1 −γ0)B θ 1 2 ∞τ´dτ. We used the bounds z1 2(s)−1>0 and τ < t. Now, taking into account formula (4.1), Ψ(t)≤ ≤ C exp³−(1−γ0)B θ 1 2 ∞t´Zt 0 exp ½·−µ3 2−p¶2Fθp−1 ∞+(1 −γ0)B θ 1 2 ∞¸τ¾dτ =C exp ¡−¡3 2−p¢2Fθp−1 ∞t¢−exp ³−(1 −γ0)B θ 1 2 ∞t´ −¡3 2−p¢2Fθp−1 ∞+ (1 −γ0)B θ 1 2 ∞ where C=F θp ∞·cµ1 2¶+c(p)¸ =F θp ∞ 3−2p(max n1, zp−1 0o+ 2 pmax n1, z2p−3 2 0o)¯¯¯z 3 2−p 0−1¯¯¯. Hence Ψ(t) decays exponentially and the exponential decay of d2is proved. Depending on the sign of −µ3 2−p¶2Fθp−1 ∞+ (1 −γ0)B θ 1 2 ∞ (which vanishes for p=1 2), we conclude that C3=(¡3 2−p¢2Fθp−1 ∞,if p > 1 2 (1 −γ0)B θ 1 2 ∞,if p < 1 2 . Case z0<1. 24 M. BISI, J. A. CARRILLO AND G. TOSCANI The proof is a bit more complicated than the previous one, because now z1 2(s)−1<0. To estimate Ψ(t), using (4.2) we recover the bound exp µ(1 −γ0)B θ 1 2 ∞Zt 0h1−z1 2(s)ids¶ ≤exp ½(1−γ0)1 (3−2p)2 B Fz−1 2 0θ 3 2−p ∞max n1, zp−1 0o³1−z 3 2−p 0´¾=: K. Precisely we have Ψ(t) = exp µ(1 −γ0)B θ 1 2 ∞Zt 0h1−z1 2(s)ids −(1 −γ0)B θ 1 2 ∞t¶ ×Zt 0 ϕ(τ) exp µ−(1 −γ0)B θ 1 2 ∞Zτ 0h1−z1 2(s)ids + (1 −γ0)B θ 1 2 ∞τ¶dτ ≤ K exp ³−(1 −γ0)B θ 1 2 ∞t´Zt 0 ϕ(τ) exp ³(1 −γ0)B θ 1 2 ∞τ´dτ. Applying again formula (4.2) to the function ϕ(τ), it follows Ψ(t)≤ CK exp ³−(1 −γ0)B θ 1 2 ∞t´ ×Zt 0 exp ½·−µ3 2−p¶2Fθp−1 ∞z 1 2 0+ (1 −γ0)B θ 1 2 ∞¸τ¾dτ =C K exp ³−¡3 2−p¢2Fθp−1 ∞z 1 2 0t´−exp ³−(1 −γ0)B θ 1 2 ∞t´ −¡3 2−p¢2Fθp−1 ∞z 1 2 0+ (1 −γ0)B θ 1 2 ∞ , where Cis the same constant obtained in the previous case. Also in this case Ψ(t) decreases at an exponential rate. Let us remark again that the previous decay holds if −µ3 2−p¶2Fθp−1 ∞z 1 2 0+ (1 −γ0)B θ 1 2 ∞ does not vanish. Depending on the sign of this factor then one obtains the value of C3similarly to the case z0>1. DIFFUSIVE EQUILIBRIA 25 Remark 4.3. Theorems 3.2–4.2 can be stated also for the inelastic one– dimensional Kac model introduced in Ref. [19]: ∂ˆ f ∂t =pθ(t) 2πZ2π 0·ˆ f(k− p)ˆ f(k+ p)−ˆ f(k)ˆ f(0)¸dϑ −F θp(t)|k|2ˆ f where k− p=ksin ϑ|sin ϑ|pk+ p=kcos ϑ|cos ϑ|p. Remark 4.4. The theorems proved in this section imply the convergence of other probability metrics that turn out to be equivalent to ds, that is metrics which define the same weak?–uniformity on the class of probability distributions having bounded the moment of order s[15, 21]. 5. Propagation of smoothness for the Boltzmann equation The goal of this section is to show that the smoothness of the initial data of the Boltzmann equation is propagated so that we have bounds on the smoothness uniform in time. Using the same computations of step 3 of Theorem 3.2, we obtain the inequality d dt ZR3 |k|2m|ˆ f(k)|2dk ≤ 2Bpθ(t)·1 4πZS2 ZR3 |k|2m¯¯¯ ˆ f(k−)ˆ f(k+)ˆ fc(k)¯¯¯dn dk −ZR3 |k|2m|ˆ f(k)|2dk¸ −2Fθ(t)pZR3 |k|2m+2|ˆ f(k)|2dk. (5.1) Let us remark that, thanks to the bound (3.13) ZR3 1 4πZS2 |k|2m¯¯¯ ˆ f(k−)ˆ f(k+)ˆ fc(k)¯¯¯dn dk ≤C(m, e)ZR3 |k|2m|ˆ f(k)|2dk, where C(m, e)>1 is a constant which depends only on mand on the coefficient of restitution e, and can be explicitly computable. Hence, if Zm(t) = kf(t)kHm,Zm(t) satisfies the inequality dZm(t) dt ≤2Bpθ(t) [C(m, e)−1] Zm(t)−2Fθ(t)pZm+1(t).(5.2) The desired result follows from (5.2) by virtue of the following Nash-type inequality. Lemma 5.1. Let f∈Hm+1(R3)be a probability density function. Then, f∈Hm(R3)and the following bound holds kfkHm+1(R3)≥cm¡kfkHm(R3)¢(2m+5)/(2m+3) (5.3)