Well-posedness and time-asymptotic of Boltzmann equations for monatomic and polyatomic mixtures

TL;DR

This paper establishes well-posedness and decay properties of Boltzmann equations modeling mixtures of monatomic and polyatomic gases, introducing novel methods to handle internal energy variables and dissimilar component masses.

Contribution

It develops a new approach to analyze internal energy variables and asymmetries in multi-species Boltzmann equations, proving well-posedness and decay rates.

Findings

Proves polynomial decay in space for the whole space case.

Establishes exponential decay in the torus case.

Provides a structure for fluid limit analysis.

Abstract

This paper considers a system of Boltzmann equations modelling the mixture of monatomic and polyatomic gases in an $L^{2}-L^{\infty}$ perturbation theory around global modified Maxwellians accounting for the internal energy of the mixture in the whole space and the torus. We investigate the pointwise decay in velocity and internal energy of the linearized Boltzmann operators in the four types of collisions. A novel approach is developed to deal with the additional internal energy variable $I\in \mathbb{R}_+$ and the loss of symmetry due to dissimilar masses of the mixture components. Subsequently, we carry out a classical $L^2-L^\infty$ method to establish the well-posedness theory of the system. The optimal polynomial time decay rate on the whole space is obtained accordingly based on the spatial Fourier's study of the linearized system. The analysis shows the structure of a perturbed Euler-type model for the solution's macroscopic quantities: density, bulk velocity, and temperature, near the steady state, which gives a potential application to investigate fluid limit problems. In addition, this work proves exponential time decay in the torus and fills the gap of classical multi-species Boltzmann in the whole space.