The weak solution to a Boltzmann type equation and its energy conservation

TL;DR

This paper establishes the existence of global weak solutions for a Boltzmann type equation with nonlinear damping in three dimensions and proves energy conservation under specific conditions.

Contribution

It introduces a novel approach using a variant Gronwall inequality and $L^p$ regularity to prove existence and energy conservation of weak solutions for a nonlinear Boltzmann type equation.

Findings

Existence of global weak solutions with large initial data

Energy conservation under certain conditions

Use of $L^p$ regularity and compactness arguments

Abstract

In this paper, we study the initial value problem of a Boltzmann type equation with a nonlinear degenerate damping. We prove the existence of global weak solutions with large initial data, in three dimensional space. We rely on a variant version of the Gronwall inequality and $L^p$ regularity of average velocities to derive the compactness of solutions to a suitable approximation. This allows us to recover a weak solution by passing to the limits. After the existence result, we also prove energy conservation for the weak solution under some certain condition.