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

TL;DR

The paper proves well-posedness and time-asymptotics for a Boltzmann system modeling a monatomic–polyatomic mixture with an internal energy variable $I$, on both $ ext{R}^3$ and $ ext{T}^3$. It develops an $L^2$–$L^ extinfty$ framework around a global modified Maxwellian, addressing the four collision types via a velocity–internal-energy splitting to handle dissimilar masses and internal energy effects. The main contributions are explicit pointwise collision-operator bounds, robust decay estimates for the linearized operator, and a nonlinear stability theory yielding global mild solutions with optimal polynomial decay in space and exponential decay on the torus. These results lay groundwork for fluid limits and hydrodynamic-type descriptions of polyatomic-monatomic gas mixtures. The work enhances the mathematical understanding of kinetic models for gas mixtures and provides tools for future bounded-domain and hydrodynamic-limit analyses.

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.

Theorems & Definitions (25)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.1
• Proposition 2.2
• proof : Proof of Theorem \ref{['global']}
• Lemma 3.1: Poly-Poly Collision
• proof
• Lemma 3.2: Mono-Poly Collision