Global bounded solutions to the Boltzmann equation for a polyatomic gas

Abstract

In this paper, we consider the Boltzmann equation modeling the motion of a polyatomic gas where the integration collision operator in comparison with the classical one involves an additional internal energy variable [Formula: see text] and a parameter [Formula: see text] standing for the number of internal degrees 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 [Formula: see text] approach. Precisely, we first study the [Formula: see text] decay property for the linearized equation, then use the iteration technique for the linear integral operator to get the linear weighted [Formula: see text] decay, and in the end obtain [Formula: see text] 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 [Formula: see text].