Infinite energy solutions to inelastic homogeneous Boltzmann equation

TL;DR

This paper investigates infinite-energy equilibrium solutions for inelastic homogeneous Boltzmann equations, establishing their existence, stability, and convergence properties in high-dimensional kinetic models.

Contribution

It introduces the existence of nontrivial stable solutions as scale mixtures of alpha-stable laws and characterizes their mixing distribution as a fixed point, extending understanding of inelastic kinetic equations.

Findings

Existence of nontrivial stationary solutions as alpha-stable mixtures.

Convergence of transient solutions to equilibrium via the central limit theorem.

Characterization of the mixing distribution as a fixed point of a smoothing transformation.

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 $d \geq 3$. Our results cover in particular Bobyl\"ev'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.