Measure Valued Solution to the Spatially Homogeneous Boltzmann Equation with Inelastic Long-Range Interactions

TL;DR

This paper establishes the well-posedness of measure-valued solutions to the inelastic Boltzmann equation with long-range interactions without angular cutoff, extending the understanding of infinite energy solutions and their asymptotic behavior.

Contribution

It introduces a framework for well-posedness of measure solutions for the inelastic Boltzmann equation without cutoff and extends self-similar solutions to the inelastic case.

Findings

Well-posedness of measure solutions established

Extension of self-similar solutions to inelastic case

Asymptotic stability of solutions demonstrated

Abstract

This paper is to study the inelastic Boltzmann equation without Grad's angular cutoff assumption, where the well-posedness theory of the solution to the initial value problem is established for the Maxwellian molecules in a space of probability measure defined by Cannone-Karch in [Comm. Pure. Appl. Math. 63 (2010), 747-778] via Fourier transform and the infinite energy solutions are not a priori excluded as well. Meanwhile, the geometric relation of the inelastic collision mechanism is introduced to handle the strong singularity of the non-cutoff collision kernel. Moreover, we extend the self-similar solution to the Boltzmann equation with infinite energy shown by Bobylev-Cercignani in [J. Stat. Phy. 106 (2002), 1039-1071] to the inelastic case by a constructive approach, which is also proved to be the large-time asymptotic steady solution with the help of asymptotic stability result in a certain sense.