Infinite energy solutions to inelastic homogeneous Boltzmann equation

Abstract

This paper is concerned with the existence, shape and dynamical stability of infinite-energy equilibria for a general class of spatially homogeneous kinetic equations in space dimensions $d\ge 3$. Our results cover in particular Bobylëv's model for inelastic Maxwell molecules. First, we show under certain conditions on the collision kernel, that there exists an index $\alpha \in (0,2)$ such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric $\alpha$-stable laws. We also characterize the mixing distribution as the fixed point of a smoothing transformation. Second, we prove that any transient solution that emerges from the NDA of some (not necessarily radial symmetric) $\alpha$-stable distribution converges to an equilibrium. The key element of the convergence proof is an application of the central limit theorem to a representation of the transient solution as a weighted sum of i.i.d. random vectors.

References in corpus (3)

• Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model
• Probabilistic View of Explosion in an Inelastic Kac Model
• Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions

Cited by in corpus (1)

• Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations