Global renormalized solutions to Boltzmann systems modeling mixture gases of monatomic and polyatomic species

TL;DR

This paper establishes the existence of global renormalized solutions for Boltzmann systems modeling mixtures of monatomic and polyatomic gases, incorporating internal energy variables, and verifies their entropy properties.

Contribution

It constructs smooth approximations, derives uniform bounds, and proves the existence of renormalized solutions satisfying entropy inequalities for complex gas mixtures.

Findings

Existence of global renormalized solutions established.

Solutions satisfy the entropy inequality.

Method employs averaged velocity-internal energy lemma.

Abstract

Inspired by DiPerna-Lions' work \cite{Diperna-Lions}, we study the renormalized solutions to the large-data Cauchy problem of the Boltzmann systems modeling mixture gases of monatomic and polyatomic species, in which the distribution functions $f_\alpha$ characterized the polyatomic species contain the continuous internal energy variable $I \in \mathbb{R}_+$. We first construct the smooth approximated problem and establish the corresponding uniform and physically natural bounds. Then, by employing the averaged velocity (-internal energy) lemma, we can show that the weak $L^1$ limit of the approximated solution is exactly a renormalized solution what we required. Moreover, we also justify that the constructed renormalized solution subjects to the entropy inequality.