Global bounded solutions to the Boltzmann equation for a polyatomic gas

TL;DR

This paper proves the global existence and exponential decay of solutions to the Boltzmann equation for polyatomic gases, incorporating internal energy variables and degrees of freedom, using an $L^2$ and $L^ abla$ approach.

Contribution

It establishes the first global well-posedness results for the polyatomic Boltzmann equation with internal energy variables in a perturbative setting.

Findings

Proved global existence of solutions near equilibrium

Demonstrated exponential decay of solutions over time

Analyzed effects of internal energy and degrees of freedom

Abstract

In this paper we consider the Boltzmann equation modelling the motion of a polyatomic gas where the integration collision operator in comparison with the classical one involves an additional internal energy variable $I\in\mathbb{R}_+$ and a parameter $\delta\geq 2$ standing for the degree of freedom. In perturbation framework, we establish the global well-posedness for bounded mild solutions near global equilibria on torus. The proof is based on the $L^2\cap L^\infty$ approach. Precisely, we first study the $L^2$ decay property for the linearized equation, then use the iteration technique for the linear integral operator to get the linear weighted $L^\infty$ decay, and in the end obtain $L^\infty$ bounds as well as exponential time decay of solutions for the nonlinear problem with the help of the Duhamel's principle. Throughout the proof, we present a careful analysis for treating the extra effect of internal energy variable $I$ and the parameter $\delta$.