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arXiv · 2210.05191

Global bounded solutions to the Boltzmann equation for a polyatomic gas

Renjun Duan·Zongguang Li
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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 $\delta$.

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Renjun Duan, Zongguang Li. 2022-10-11. Global bounded solutions to the Boltzmann equation for a polyatomic gas. https://doi.org/10.1142/s0129167x23500362

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