One dimensional dissipative Boltzmann equation: measure solutions, cooling rate and self-similar profile

TL;DR

This paper studies the one-dimensional dissipative Boltzmann equation, establishing the optimal cooling rate, existence and uniqueness of measure solutions, and the existence of self-similar profiles using measure-theoretic and fixed point methods.

Contribution

It provides new results on the cooling rate, measure solutions, and self-similar profiles for the one-dimensional dissipative Boltzmann equation with variable hard-spheres kernel.

Findings

Optimal cooling rate determined

Existence and uniqueness of measure solutions proved

Self-similar profile (homogeneous cooling state) established

Abstract

This manuscript investigates the following aspects of the one dimensional dissipative Boltzmann equation associated to variable hard-spheres kernel: (1) we show the optimal cooling rate of the model by a careful study of the system satisfied by the solution's moments, (2) give existence and uniqueness of measure solutions, and (3) prove the existence of a non-trivial self-similar profile, i.e. homogeneous cooling state, after appropriate scaling of the equation. The latter issue is based on compactness tools in the set of Borel measures. More specifically, we apply a dynamical fixed point theorem on a suitable stable set, for the model dynamics, of Borel measures.