Generalized Kinetic Maxwell Type Models of Granular Gases
In this chapter we consider generalizations of kinetic granular gas models given by Boltzmann equations of Maxwell type. These type of models for nonlinear elastic or inelastic interactions, have many applications in physics, dynamics of granular gases, e
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Summary. In this chapter we consider generalizations of kinetic granular gas models given by Boltzmann equations of Maxwell type. These type of models for nonlinear elastic or inelastic interactions, have many applications in physics, dynamics of granular gases, economy, etc. We present the problem and develop its form in the space of characteristic functions, i.e., Fourier transforms of probability measures, from a very general point of view, including those with arbitrary polynomial nonlinearities and in any dimension space. We find a whole class of generalized Maxwell models that satisfy properties that characterize the existence and asymptotic of dynamically scaled or self-similar solutions, often referred as homogeneous cooling states. Of particular interest is a concept interpreted as an operator generalization of usual Lipschitz conditions which allows to describe the behavior of solutions to the corresponding initial value problem. In particular, we present, in the most general case, existence of self similar solutions and study, in the sense of probability measures, the convergence of dynamically scaled solutions associated with the Cauchy problem to those self-similar solutions, as time goes to infinity. In addition we show that the properties of these self-similar solutions lead to non classical equilibrium stable states exhibiting power tails. These results apply to 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 presentation.
1 Introduction It has been noticed in recent years that a significant non-trivial physical phenomena in granular gases can be described mathematically by dissipative Boltzmann type equations, as can be seen in [17] for a review in the area. As motivated by this particular phenomena of energy dissipation at the kinetic level, we consider in this chapter the Boltzmann equation for non-linear interactions of Maxwell type and some generalizations of such models.
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A.V. Bobylev et al.
The classical conservative (elastic) Boltzmann equation with the Maxwelltype interactions is well-studied in the literature (see [5, 14] and references therein). Roughly speaking, this is a mathematical model of a rarefied gas with binary collisions such that the collision frequency is independent of the velocities of colliding particles, and even though the intermolecular potentials are not of those corresponding to hard sphere interactions, still these models provide a very rich inside to the understanding of kinetic evolution of gases. Recently, Boltzmann equations of Maxwell type were introduced for models of granular gases were introduced in [7] in three dimensions, and a bit earlier in [3] for in one dimension case. Soon after that, these models became very popular among the community studying granular gases (see, for example, the book [13] and references therein). There are two obvious reasons for such studies The first one is
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