Asymptotic Dynamics for Generalized Non-linear Kinetic Maxwell-type Models

Maxwell-type models for non-linear kinetic equations have many applications in physics, dynamics of granular gases, economics, and other areas where statistical modeling is relevant. They model the evolution of probability measures (distribution functions) and relate to a class of non-linear space homogeneous Boltzmann equations for binary interactions where scattering probability rates of the two particles at the time of the interaction are independent of their relative velocity. In this lecture, I have considered a large class of generalized multi-linear interacting models of Maxwell type from a rather general viewpoint, including those with arbitrary polynomial non-linearities and in any dimension space. By working in Fourier space (i.e., the space of its associated characteristic functions), it is possible to show that this class of generalized Maxwell models satisfies three fundamental properties that allow us to describe in detail the behavior of solutions depending on their initial states. In particular, it is possible to 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 self-similar solutions, as time goes to infinity. The properties of these self-similar solutions, which lead to Non (classical) Equilibrium Stable States (NESS), are studied in detail in [8]. More specifically, the classical elastic Boltzmann equation with Maxwelltype interactions, a mathematical model of a rarefied gas with binary collisions such that the collision frequency is independent of the velocities of colliding particles, has been well-studied in the literature (see [4], [13] and references therein). It is also well established that, due to micro reversibility (elastic, energy conservative) interactions, the model satisfies the Boltzmann H-theorem which yields the long time convergence of any solution of the space homogeneous initial value problem with finite energy to the Maxwell-Boltzmann probability distribution. Maxwell-type models for granular gases were introduced relatively recently in [5] in the mathematical physics framework (see also [2] for the