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99
Moment inequalities and highenergy tails for Boltzmann equations wiht inelastic interactions
 J. Stat. Phys
, 2004
"... Abstract. We study the highenergy asymptotics of the steady velocity distributions for model systems of granular media in various regimes. The main results obtained are integral estimates of solutions of the hardsphere Boltzmann equations, which imply that the velocity distribution functions f(v) ..."
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Cited by 61 (9 self)
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Abstract. We study the highenergy asymptotics of the steady velocity distributions for model systems of granular media in various regimes. The main results obtained are integral estimates of solutions of the hardsphere Boltzmann equations, which imply that the velocity distribution functions f(v) behave in a certain sense as C exp(−rv  s) for v  large. The values of s, which we call the orders of tails, range from s = 1 to s = 2, depending on the model of external forcing. The method we use is based on the moment inequalities and careful estimating of constants in the integral form of the Povznertype inequalities.
LongTime Asymptotics of Kinetic Models of Granular Flows
 Arch. Rational Mech. Anal
, 2003
"... We analyze the longtime asymptotics of certain onedimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coe#cient of restitution. These nonlinear ..."
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Cited by 46 (6 self)
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We analyze the longtime asymptotics of certain onedimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coe#cient of restitution. These nonlinear equations, classified as nonlinear friction equations, split naturally into two classes, depending whether their similarity solutions (homogeneous cooling state) extinguish or not in finite time. For both classes, we show uniqueness of the solution by proving decay to zero in the Wasserstein metric of any two solutions with the same mass and mean velocity. Furthermore, if the similarity solution extinguishes in finite time, we prove that any other solution with initially bounded support extinguishes in finite time, by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support.
Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy Problem
"... We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity. It covers physically realistic mode ..."
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Cited by 45 (13 self)
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We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity. It covers physically realistic models for granular materials. We prove (local in time) nonconcentration estimates in Orlicz spaces, from which we deduce weak stability and existence theorem. Strong stability together with uniqueness is proved under additional smoothness assumption on the initial datum, for a restricted class of collision rates. Concerning the longtime behaviour, we give conditions for the cooling process to occur or not in finite time.
On the Boltzmann equation for diffusively excited granular media
 Comm. Math. Phys
"... Abstract. We study the Boltzmann equation for a spacehomogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L 2 (R N) function, with bounded mass and kinetic energy (second moment), we p ..."
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Cited by 44 (5 self)
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Abstract. We study the Boltzmann equation for a spacehomogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L 2 (R N) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the highvelocity tails of both the stationary and timedependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions.
Asymptotic properties of the inelastic Kac model
 J. Statist. Phys
, 2004
"... Abstract We discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of Kac caricature of a Maxwellian gas. It is shown that global equilibria different from concentration are possible if the energy is not finite. These equilibria are distribute ..."
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Cited by 37 (12 self)
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Abstract We discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of Kac caricature of a Maxwellian gas. It is shown that global equilibria different from concentration are possible if the energy is not finite. These equilibria are distributed like stable laws, and attract initial densities which belong to the normal domain of attraction. If the initial density is assumed of finite energy, with higher moments bounded, it is shown that the solution converges for largetime to a profile with power law tails. These tails are heavily dependent of the collision rule.
On the selfsimilar asymptotics for generalized nonlinear kinetic Maxwell models
"... Abstract. Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economy, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial nonlinearities and in any dimension sp ..."
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Cited by 26 (5 self)
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Abstract. Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economy, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial nonlinearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties which one of them can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those selfsimilar solutions, as time goes to infinity. The properties of these selfsimilar solutions, leading to non classical equilibrium stable states, are studied in detail. We apply the results to three different specific problems related to the Boltzmann equation (with elastic and inelastic interactions) and show that all physically relevant properties of solutions follow directly from the general theory developed in this paper. Contents
Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails
"... Abstract. We consider the Boltzmann equations for mixtures of Maxwell gases. It is shown that in certain limiting case the equations admit selfsimilar solutions that can be constructed in explicit form. More precisely, the solutions have simple explicit integral representations. The most interestin ..."
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Cited by 25 (7 self)
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Abstract. We consider the Boltzmann equations for mixtures of Maxwell gases. It is shown that in certain limiting case the equations admit selfsimilar solutions that can be constructed in explicit form. More precisely, the solutions have simple explicit integral representations. The most interesting solutions have finite energy and power like tails. This shows that power like tails can appear not just for granular particles (Maxwell models are far from reality in this case), but also in the system of particles interacting in accordance with laws of classical mechanics. In addition, nonexistence of positive selfsimilar solutions with finite moments of any order is proven for a wide class of Maxwell models. To Carlo Cercignani, on his 65th Birthday 1.
On a kinetic model for a simple market economy
, 2004
"... In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasistationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of ..."
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Cited by 20 (8 self)
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In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasistationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a FokkerPlanck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis.
Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria
 J. Stat. Phys
"... We quantify the longtime 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 bas ..."
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Cited by 19 (8 self)
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We quantify the longtime 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 L 1norm, as well as various Sobolev norms. 1