Global bounded solutions to the Boltzmann equation for a polyatomic gas

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 \(\delta\).