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arXiv · 1309.7217

Infinite energy solutions to inelastic homogeneous Boltzmann equation

Federico Bassetti·Lucia Ladelli·Daniel Matthes
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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 Bobylëv's model for inelastic Maxwell molecules. First, we show under certain conditions on the collision kernel, that there exists an index $α\in(0,2)$ such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric $α$-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) $α$-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.

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Federico Bassetti, Lucia Ladelli, Daniel Matthes. 2013-09-27. Infinite energy solutions to inelastic homogeneous Boltzmann equation. https://arxiv.org/abs/1309.7217

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