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# Global bounded solutions to the Boltzmann equation for a polyatomic gas

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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 $$\delta$$.

### Author and article information

###### Journal
11 October 2022
###### Article
2210.05191
96790b77-c078-4dbc-9660-974a78c0c074