probabilistic treatment of some dissipative kinetic models

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of finite energy. These equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction. We remark in this.
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PROBABILISTIC TREATMENT OF SOME DISSIPATIVE KINETIC MODELS∗

A. PULVIRENTI AND G.TOSCANI Department of Mathematics University of Pavia via Ferrata 1, 27100 Pavia, ITALY E-mail: [email protected]; [email protected]

We present and discuss here some one–dimensional kinetic models with dissipative interactions. The first model is obtained from a suitable modification of Kac caricature of a Maxwellian gas, while the second, a fractional Fokker–Planck type equation, is obtained from the modified Kac equation with a singular kernel, in the so–called grazing collision limit. It is shown that both models admit global equilibria different from concentration, provided that we leave the usual assumption of finite energy. These equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction. We remark in this way the connections between the large–time behavior of dissipative models and the classical central limit theorem for stable laws. These connections enable us to make use of arguments typical of probability theory to recover the rate of decay towards equilibrium.

Key words. Granular gases, Boltzmann-like dissipative equations, long-time behavior of solutions. AMS(MOS) subject classification. 76P05, 82C40.

1. Introduction As it has been noticed by many authors, starting with the pioneering papers by Kac and McKean17,18 , there are strong analogies between the central limit theorem of probability theory and the trend to equilibrium for the spatially homogeneous Boltzmann equation. These analogies are almost evident in the case of Maxwellian interactions, because in this case there is a clear similarity between ∗ Work

partially supported both by the project HYKE, “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis” financed by the European Union (IHP), Contract Number HPRN-CT-2002-00282, and by the Italian MIUR project “Mathematical Problems of Kinetic Theories”. 1

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the gain term and a classical convolution operator, which suggests to make use of some arguments typical of the proofs of the central limit theorem in the context of the Boltzmann equation13,25 . In a recent paper, Bobylev and Cercignani7 considered a class of solutions of the elastic Boltzmann equation (for Maxwellian molecules) with infinite energy, discussing briefly the question of identifying the asymptotic behavior of the Boltzmann equation in this case. They remarked that in probability theory, the analogous problem is the case of infinite variance. Under this condition, the right scaling is not, like in the classical central limit theorem √ (X1 + ... + Xn )/ n, (1) but

(X1 + ... + Xn )/n1/s

(2)

for some exponent 0 < s < 2 which depends on the tail behavior of the distribution function of the independent random variables Xn 16,15 . Then, the convergence is not towards a Gaussian distribution, but towards one of the so–called stable laws. Among them, the simplest one are known as Levy laws, and in the symmetric case their Fourier transforms look like exp{−α|ξ|β } for some constants α > 0 and 0 < β < 2. A further remark shows that their idea can be fruitfully applied to the Boltzmann equation with dissipative collisions. In fact, a direct inspection of (2) shows that, if the variance of the random variables Xn is finite, the variance of the sum rescaled according to n1/s , 0 < s < 2, is decreasing with respect to n. Since the energy in the Boltzmann equation is decreasing when the binary collisions are dissipative, it is immediate to conclude that, in the case of stable laws, the analogy between the central limit theorem and the Boltzmann equation will be stronger if, in addition to the requirement of unbounded energy, we consider the inelastic Boltzmann equation. In recent times the study of the free and driven cooling of dissipative granular gases has received a lot of attention. The investigation of simplified models made it possible to achieve essential progresses and to verify or discard conjectures. In particular, a very fruitful research field of activity was the application of the Boltzmann equation to inelastic Maxwell particles for both the free case without energy input5,3,11,12 and the driven one9,6 . Inelastic Maxwell models share with elastic Maxwell molecules the property that the collision rate in the Boltzmann equation is independent of the relative velocity of the colliding pair. Even if these models do not describe real particles, they are of interest for granular fluids in spatially homogeneous states because of the mathematical simplifications resulting from a velocity independent collision rate. Differently from elastic collisions, partially inelastic collisions have a nontrivial outcome also in dimension one, so that the one-dimensional idealization turns out to provide interesting models24 . Among others, a one–dimensional pseudo– Maxwellian inelastic gas was studied recently1 , and an exact similarity solution was found. A one-dimensional dissipative kinetic model which is sufficiently rich to exhibit a variety of steady states and similarity solutions was introduced and discussed

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by the authors21 . There, the existence of non Maxwellian equilibria with infinite energy which are Levy distributions can be shown. This enlightens the analogies between the central limit theorem for stable laws and the convergence towards equilibria with infinite energy in dissipative gases, as the classical central limit theorem is closely connected with the convergence towards equilibrium of the elastic Boltzmann or Kac equation. Having in mind these analogies, one can obtain both uniqueness and convergence results in terms of a distance which has been used recently in connection with the Boltzmann equation13,8,25 and the central limit theorem for stable laws14 as well. The main result is that these Maxwellian equilibria attract any initial data which is in a suitable (small) domain of attraction. The last part of the paper is devoted to the discussion of the problem of the grazing collision limit26 of the dissipative Kac equation in the case of solutions with unbounded energy. We recover in this way a fractional type Fokker-Planck equation introduced recently in the literature in connection with stochastic collision models19,2 . 2. Kinetic aspects of the central limit theorem for stable laws The aim of this section is to focus on the analogies between the convergence towards a stable law in the central limit theorem15,16 and the large-time asymptotic of some dissipative kinetic models. The central limit theorem for a centered s–stable law with distribution θ consists of finding the set of distributions F such that, when dealing with the normalized sum X + ... + Xn Sn = 1 1/s n of the independent and identically distributed random variables Xi with common distribution function F , the distribution Fn of the sum Sn converges to θ. Let us denote with f (x), x ∈ R, the probability density function of the random variables Xi , where f is normalized (with zero expectation and unit variance), and with fn (x) the density of S2n , n ≥ 1. Then, since S2n+1 =

1 1 S2n + 1/s S2∗n , 21/s 2

where S2n and S2∗n are independent and identically distributed, Z ³ ´ ³ ´ fn+1 (x) = dy 21/s fn 21/s (x − y) 21/s fn 21/s y .

(3)

R

Let us change variable into the integral in (3), setting x − y = 12 (x + z) (which implies y = 21 (x − z)). We obtain µ ¶ µ ¶ Z 2−s x+z x−z dz 2 s fn f . (4) fn+1 (x) = n s−1 s−1 R 2 s 2 s Since fn (x) has unit mass, we can rewrite (3) as ½ µ ¶ µ ¶ ¾ Z 2−s x+z x−z dz 2 s fn fn+1 (x) = fn (x) + f − f (x)f (z) n n n s−1 s−1 R 2 s 2 s

(5)

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The recursive relation (5) can be viewed as the explicit Euler scheme (at discrete times ∆t = 1) of the kinetic equation ∂f (x, t) = Q∗s (f, f )(x, t), ∂t where

Z Q∗s (f, f )(x) =

n dz

2

2−s s

R

o ¡ ¢ ¡ ∗∗ ¢ f x∗∗ f zs − f (x)f (z) . s

(6)

(7)

∗∗ In (7) (x∗∗ s , zs ) are the pre-collisional velocities in the collision having (x, z) as post-collisional velocities. Hence, the post-collisional velocities (x∗s , zs∗ ) are related to (x, z) by the relations

x∗s = zs∗

=

1 2s (x + z) 1 2s (x − z).

Since for s < 2 (x∗s )2 + (zs∗ )2 = 2

s−2 s

³

(8) (9)

´ x2 + z 2 < x2 + z 2 ,

the operator dissipates the second moment (energy), while for s = 2 the energy is conserved. This clarifies the deep analogies between the classical problem of the convergence towards a stable law, and the cooling problem in kinetic theory of dissipative gases. From this analogy it follows at once that methods developed for the former problem can be fruitfully applied to the latter. 3. The inelastic Kac model To clarify the analogy between the kinetic–like equation for densities in the central limit theorem for stable laws, and the cooling problem for a gas with dissipative collisions, we shall introduce here a suitable generalization of the Kac caricature of a Maxwellian gas17 , which allows to take into account dissipative effects. To this end, we consider the equation21 ∂f (v, t) = Qp (f, f )(v, t) ∂t

(10)

where the right-hand side of (10) describes the rate of change of the density function f due to dissipative collisions, Z h i dθ Qp (f, f )(v) = dw χ−1 f (vp∗∗ )f (wp∗∗ ) − f (v)f (w) . (11) R×[−π,π] 2π The velocities (vp∗∗ , wp∗∗ ) are the pre collisional velocities of the so–called inverse collision, which results with (v, w) as post collisional velocities. Given (v, w), the post collisional velocities (vp∗ , wp∗ ) are defined simply generalizing the Kac rule vp∗ = v cos θ| cos θ|p − w sin θ| sin θ|p wp∗ = v sin θ| sin θ|p + v cos θ| cos θ|p .

(12)

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In (12) the positive constant p < +∞ measures the degree of inelasticity. If p = 0, the binary collision is elastic and we obtain the classical Kac equation, where the post collisional velocities are given by a rotation in the (v, w) plane. The factor χ = | sin θ|2+2p + | cos θ|2+2p in the gain term appears respectively from the Jacobian of the transformation dv ∗∗ dw∗∗ into dvdw. The lost of energy in a single binary collision depends on the choice of the inelasticity parameter p, and it is given by ³ ´ (vp∗ )2 + (wp∗ )2 = (v 2 + w2 ) | sin θ|2+2p + | cos θ|2+2p . (13) The structure of the inelastic Kac equation is similar to the inelastic Boltzmann equation for a Maxwell gas, and also here mass is conserved, while energy is nonincreasing. However, momentum is not conserved unless it is zero initially, and therefore this will be the only case considered. Moreover, the Kac equation does not obey an H-theorem. Here normalizations are chosen in such a way that Z Z Z f (v, 0) dv = 1 ; vf (v, 0) dv = 0 ; v 2 f (v, 0) dv = 1. (14) R

R

R

One of the important properties of Maxwell models is that the moment equations form a set of closed equations. Indeed, given any function ϕ = ϕ(v), the evolution of hϕi is given by Z d ϕ(v)f (v, t) dv = dt R Z £ ¤ 1 dθ dv dw ϕ(vp∗ ) + ϕ(wp∗ ) − ϕ(v) − ϕ(w) f (v)f (w) . (15) 2 R2 ×[−π,π] 2π Choosing ϕ(v) = v 2 shows that the average kinetic energy of granular temperature T (t) = hv 2 i keeps decreasing at a rate proportional to the inelasticity Z π ´ dθ ³ γp = 1 − | sin θ|2+2p + | cos θ|2+2p . (16) −π 2π Thus, if the initial density has finite temperature, the solution to the inelastic Kac equation does not reach the Maxwellian ω(v) = (2π)−1/2 exp[−v 2 /2], but is approaching a Dirac delta function δ(v) for large times. 4. Steady states of infinite energy The Kac model (10) can be studied in weak form Z Z ª d dθ © ϕ(vp∗ ) − ϕ(v) f (v) f (w) dv dw. ϕ(v) f (v) dv = dt 2π R×R×[−π,π] We shall study this with the normalization conditions (14). It is equivalent to use the Fourier transform of the equation4 : ³ ´ ∂ fb(ξ, t) b p fb, fb (ξ, t), =Q ∂t

(17)

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where fb(ξ, t) is the Fourier transform of f (x, t), Z fb(ξ, t) = e−iξv f (v, t) dv, R

and

In (18)

Z ³ ´ b p fb, fb (ξ) = Q

π −π

i dθ h b + b − f (ξp )f (ξp ) − fb(ξ)fb(0) . 2π

(18)

 + p  ξp = ξ cos θ| cos θ| 

ξp− = ξ sin θ| sin θ|p ,

(19)

and the initial conditions (14) turn into fb(0) = 1, fb0 (0) = 0, fb00 (0) = −1, fb ∈ C 2 (R). Hence equation (17) can be rewritten as Z π ∂ fb(ξ, t) dθ b + b − + fb(ξ, t) = f (ξp )f (ξp ). ∂t 2π −π Note that

|ξp+ |2/(1+p) + |ξp− |2/(1+p) = |ξ|2/(1+p) .

(20)

(21)

If we leave the usual hypothesis of solutions having finite energy, it is immediate to recognize that the dissipative Kac equation admits nontrivial steady states. In fact, in consequence of equality (21), the collision operator (18) vanishes if f (v) = Mp (v), where n o 2 cp (ξ) = exp −σ|ξ| 1+p , σ > 0. M (22) The function (22) is nothing but the characteristic function of a stable distribution15,16 of exponent s = 2/(1 + p), where 0 < s < 2. It is well–known that a Maxwellian is a strictly stable distribution (corresponding to p = 0), while the Cauchy law is 1–stable16 . This enlightens a remarkable difference between the problem of convergence towards equilibrium for the elastic and the inelastic Kac models, the former being the analogous of the classical central limit theorem, while the latter is the analogous of the central limit theorem for stable laws. Having in mind this analogy, it becomes clear that, while in the elastic case convergence towards the Maxwellian is proven under weak assumptions on the initial data (essentially finite moments of order strictly greater than two)13 , in the inelastic case convergence towards the steady state can be proven only if the initial data belong to the so-called normal domain of attraction15,16 . In particular, the rate of convergence towards a stable law in the central limit theorem has been recently investigated by means of a probability metric which allows for explicit computations14 . The importance of this metric in the analysis of the convergence towards equilibrium for the elastic Kac equation has been remarked in13 . We introduce and discuss this metric in the next section.

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5. Existence and uniqueness of solutions Denote by P(R) the class of all probability distributions, and with Ps (R), s > 0, the class of all probability distributions F ∈ P such that Z |v|s dF (v) < ∞. R

We introduce a metric on Ps (Rd ) by ds (F, G) = sup

ξ∈R

|fb(ξ) − gb(ξ)| |ξ|s

(23)

where fb is the Fourier transform of F , Z fb(ξ) = e−iξv dF (v). R

Let us write s = m + α, where m is an integer and 0 ≤ α < 1. In order that ds (F, G) be finite, it suffices that F and G have the same moments up to order m. The metric (23) has been introduced in13 to investigate the trend to equilibrium of the solutions to the Boltzmann equation for Maxwell molecules. There, the case s = 2 + α, α > 0, was considered. Further applications of ds , with s = 4, were subsequently studied8 , while the cases s = 2 and s = 2+α, α > 0, have been considered in connection with the so-called McKean graphs9,18 . The case s = 2 was then used25 in connection with the uniqueness of the non cut-off Boltzmann equation for Maxwell molecules. Finally, the case 0 < s < 2 led to the finding of Berry–Essen type bounds in the central limit theorem for a stable law14 . Here we shall be interested mainly in this last case, which presents several analogies with the problem under study. The existence of a solution to equation (10) can be seen easily using the same methods available for the elastic Kac model. In particular, a solution can be expressed as a Wild sum4,8 . Let f and g be two solutions of the Kac equation (10), and fb, gb their Fourier transforms. Then, given any positive constant s, ³ ´ Z π b + − + g (ξp− ) fb(ξ) − gb(ξ) ∂ f − gb dθ fb(ξp )fb(ξp ) − gb(ξp )b + = . (24) s s s ∂t |ξ| |ξ| |ξ| −π 2π Now, since ¯ ¯ ¯Ã ¯ ! ¯b ¯ − s + s ¯ fb(ξ + )fb(ξ − ) − gb(ξ + )b − ¯ ¯ f − gb ¯ |ξp | + |ξp | ¯ p p p g (ξp ) ¯ ¯ ≤ sup ¯ ¯ ¯ ¯ |ξ|s ¯ ¯ ¯ |ξ|s |ξ|s Owing to condition (21), if we choose s = 2/(1 + p) we obtain ¯ ¯ ¯ ¯ ¯ b ¯ ¯ fb(ξ + )fb(ξ − ) − gb(ξ + )b − ¯ ¯ f − gb ¯ ¯ p p p g (ξp ) ¯ ≤ sup ¯ ¯ ¯. ¯ ¯ |ξ|2/(1+p) ¯ ¯ ¯ |ξ|2/(1+p)

(25)

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We set

fb(ξ) − gb(ξ) . |ξ|2/(1+p) The computation above shows that h(t, ξ) =

∂h ≤ (khk∞ − h) . ∂t

(26)

Gronwall’s lemma proves at once that kh(t)k∞ is non increasing. We have Theorem 5.1. Let f (t) and g(t) be two solutions of the Kac equation (10), Then, if s ≥ 2/(1 + p), for all times t ≥ 0, ds (f (t), g(t)) ≤ ds (f (0), g(0)). In particular, let f0 be a nonnegative density with finite moment of order s = 2/(1 + p). Then, there exists a unique weak solution f (t) of the Kac equation, such that f (0) = f0 . 6. Central limit theorem for inelastic Kac model In the central limit problem, the normal domain of attraction (N DA) of a s– stable law consists of the set of distributions F such that, when dealing with the normalized sum X + ... + Xn Sn = 1 1/s n of the independent and identically distributed random variables Xi with common distribution function F , Sn converges to Ms . Hence, the normal domain of attraction is characterized by requiring that, for all ξ ∈ N , ¡ ¢ lim n fb(ξ/n1/s ) − 1 = −σ|ξ|s . n→∞

Equivalent definitions of the N DA in terms of properties of the distribution function can be found in16 . As in the classical problem for a normal law, information on the rate of convergence and restriction of the set of N DA (in order to get a rate of convergence) depend on the choice of the metric. In terms of the Fourier-based metric defined by (23), it is natural to introduce the set ψf (ξ) Dσ,s = {f ∈ P, fb(ξ) = 1 − σ|ξ|s + ψf (ξ), with ∈ L∞ (R)}. |ξ|s

(27)

The importance of such a set for the convergence towards equilibrium for the inelastic Kac model is easily understood if one realizes that in the elastic case Dσ,2 contains all densities with bounded second moment13 , with a precise control of the remainder ψf , which has to decay faster than |ξ|2 . It is clear that the ds (f, g) is finite when f and g belong to the set Dσ,s . Notice also that the stable steady state Ms belongs to Dσ,s (and actually the remainder goes to 0 faster than |ξ|s ).

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Following14 , the convergence problem towards equilibrium, will be discussed by introducing the Fourier domain of attraction (F DA) o n ψ(ξ) 0 −−−→ 0 . (28) Dσ,s = f ∈ Dσ,s , s |ξ| ξ→0 0 and a suitable subset of the set Dσ,p . To this extent, we consider, for δ > 0, the set 0 δ , |ψ(ξ)|/|ξ|s ≤ |ξ|δ }. = {f ∈ Dσ,s Dσ,s s

δ cs (ξ) = e−σ|ξ| and f ∈ Dσ,s Notice that, with M , one has ds+δ (f, Ms ) ≤ C < ∞. δ A easy-to-check criterion ensuring that a given density f lies in Dσ,s , relies on finiteness of some pseudo-moment. For the normal case s = 2, Ms has all moments finite, so that integrability of xα (f − Ms ) is equivalent to integrability for xα f . This of course does not apply when s < 2. Note also that, for s = 2, the set considered in13 coincides with Z δ {f ∈ P, |v|2+δ f (v) dv < ∞} ⊂ Dσ,2

with a strict inclusion. We report here the following result obtained in14 . Lemma 6.1. Let 0 ≤ s < 1 and 0 < δ ≤ 1 − s. Then, the following embedding holds Z δ {f ∈ P, |v|s+δ |f (v) − Ms (v)| dv < ∞} ⊂ Dσ,s . Let 1 ≤ s < 2 and 0 ≤ δ ≤ 2 − s. Then Z Z δ {f ∈ P, |v|s+δ |f (v) − Ms (v)| dv < ∞, v (f (v) − Ms (v)) dv = 0} ⊂ Dσ,s . We prove the main result of this section. Theorem 6.1. Let p > 1, and let f (t) be the unique solution of the Kac equation (10), corresponding to the initial density f0 such that, for some 0 < δ ≤ p−1 p+1 Z |v|s+δ |f0 (v) − Ms (v)| dv < ∞, where s = 2/(1 + p). Then, f (t) converges exponentially fast in Fourier metric towards equilibrium, and the following bound holds © ª ds+δ (f (t), Ms ) ≤ ds+δ (f0 , Ms ) exp −(1 − Ap,δ )t (29) where

Z Ap,δ =

π −π

´ dθ ³ | sin θ|2+(1+p)δ + | cos θ|2+(1+p)δ < 1 2π

Let now 0 < p ≤ 1, and let f (t) be the unique solution of the Kac equation (10), 2p corresponding to the initial density f0 such that, for some 0 < δ ≤ p+1 Z |v|s+δ |f0 (v) − Ms (v)| dv < ∞.

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Then, f (t) converges exponentially fast in Fourier metric towards equilibrium, and bound (29) holds. Proof The proof is an easy consequence of the computations leading to Theorem 5.1. Thanks to Lemma 6.1, the hypotheses on the initial value f0 are such that δ fb0 ∈ Dσ,s , so that ds+δ (f0 , Ms ) is bounded. Given δ > 0, proceeding as before we get the bound ¯ ¯ ¯ ¯Ã ! − ¯ ¯ fb(ξ + )fb(ξ − ) − gb(ξ + )b ¯ b ¯ − s+δ + |ξp+ |s+δ ¯ f − gb ¯ |ξp | ¯ p p p g (ξp ) ¯ . (30) ¯ ¯ ≤ sup ¯ s+δ ¯ ¯ ¯ ¯ |ξ| ¯ |ξ|s+δ |ξ|s+δ Using the definitions of ξp− , ξp+ , one obtains |ξp− |s+δ + |ξp+ |s+δ = | sin θ|2+(1+p)δ + | cos θ|2+(1+p)δ . |ξ|s+δ Hence, from (24) follows ´ ¯ ³ ¯ ° ° ¯ ¯ °b ° b(ξ) − gb(ξ) ¯ ¯ ∂ fb − gb f ° f (ξ) − gb(ξ) ° ¯ ¯ ≤ A ° ° p,δ ¯ ∂t |ξ|s+δ + ¯ ° |ξ|s+δ ° |ξ|s+δ ¯ ¯

(31) ∞

Gronwall’s lemma then proves (29).

7. Grazing collision limit and the fractional Fokker–Planck equation The non cut-off Kac equation was introduced by Desvillettes10 who showed that the presence of the singularity in the kernel introduces a gain of regularity in the solution, and that this gain is related to the number of moments that are initially finite. Subsequently, the grazing collision limit26 for this equation was studied, and the convergence of the collision operator to the linear Fokker–Planck operator was proven23,20 . The method of proof can be adapted to the inelastic Kac operator, to study the grazing collision asymptotics of the model, when the kernel is singular22 . The inelastic-Kac operator with singular kernel β(θ) reads Z h i Qβ (f, f )(v) = β(θ) dw χ−1 f (vp∗∗ )f (wp∗∗ ) − f (v)f (w) . (32) p π R×[− π 2,2]

The kernel β(θ) has a non-integrable singularity for θ = 0, and ∂f (v, t) = Qβ (33) p (f, f )(v, t) ∂t is a reasonable model to study the grazing-collision limit for inelastic collisions. Without getting into the details of the analysis, the grazing limit is obtained in this case by taking a sequence of scattering cross sections which depend on a small parameter ² and concentrate to zero as ² tends to zero in such a way that Z π 2 β² (θ) sin2 θd θ → 1. (34) −π 2

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This condition is different from the analogous one considered in the elastic model, which is Z π 2

−π 2

β(θ) sin θd θ → 1.

(35)

This reflects the fact that in the elastic case during the grazing collision process we assume that the momentum transfer (which is linear in the parameter θ) remains bounded, while in the dissipative case we assume bounded the dissipation of energy (which is quadratic in θ). One obtains22 Theorem 7.1. Let p > 1, and let f² (t) be the unique solution of the Kac equation (33), with kernel β² , corresponding to the initial density f0 belonging to the Fourier domain of attraction (28) for s = 2/(1 + p), and let us set lim

ξ→0+

fb0 (ξ) − fb0 (0) |ξ|

2 1+p

− lim

ξ→0−

fb0 (ξ) − fb0 (0) 2

|ξ| 1+p

= 2D(f0 )

(36)

Then, if the sequence of kernels satisfies (34), f² (t) converges to f (t), solution of the fractional Fokker-Planck equation ∂f (v, t) ∂ 2/(1+p) f (v, t) ∂ = 2D(f0 ) + (1 + p) [(vf (v, t)] , ∂t ∂v ∂|v|2/(1+p)

(37)

where ∂ α /∂|v|α is a Riesz fractional derivative. By using again the Fourier transform of the equation, one is led to study the behavior of the sequence {fb² } of solutions to ³ ´ ∂ fb(ξ, t) ² bβ =Q fb, fb (ξ, t), p ∂t with

Z ³ ´ ² b b bβ Q f , f (ξ) = p

π 2

−π 2

h i β² (θ) fb(ξp+ )fb(ξp− ) − fb(ξ)fb(0) d θ.

(38)

(39)

Assume now that the initial datum f0 satisfies condition (36). Then, using the Taylor expansions of fb(ξp+ ) and fb(ξp− ), as well as (34) it is possible to prove22 that the solution fb² (ξ), converges to fb(ξ), where fb satisfies the limit equation 2 ∂ fb ∂ fb = −2D|ξ| 1+p fb(ξ) − (1 + p)ξ ∂t ∂ξ

(40)

Equation (40) is known as fractional Fokker-Planck equation2,19 and can be considered as an approximation of the inelastic Kac collision operator when the collisions become grazing. The first term on the right hand side describes, through its coefficient 2D, fractional diffusion in velocity space, while the second term describes dissipation through the factor 1 + p. Note that D(f0 ) is zero when the energy of the initial datum is finite, and in this case the right–hand side of equation (37) reduces to a linear friction.

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