Boltzmann framework for polyatomic gases: review on well-posedness, higher integrability and physical relevance

TL;DR

This review discusses the mathematical analysis of the scalar Boltzmann equation for polyatomic gases, focusing on well-posedness, integrability, and physical relevance of the collision kernel, supported by comparison with experimental data.

Contribution

It provides a comprehensive review of well-posedness, integrability, and physical relevance results for the scalar Boltzmann equation for polyatomic gases.

Findings

Established $L^1$-theory for homogeneous solutions with positive mass and bounded energy.

Derived entropy-based estimates and $L^p$-integrability properties of solutions.

Validated the physical relevance of the collision kernel through comparison with experimental data.

Abstract

This paper reviews results on the scalar Boltzmann equation for a single-component polyatomic gas with continuous internal energy. For the space homogeneous problem, $L^1$-theory is established, for solutions with initial strictly positive mass and bounded energy, which enables to solve the Cauchy problem for initial data with $L^1_{2^+}$-moments using the comparison principle for ODEs. Then, deriving entropy-based estimates, $L^p$-integrability properties of the solution are explored, $p\in (1,\infty]$. All these analytical results hold under a specific assumption on the collision kernel corresponding to cut-off and hard-potentials type. A mean to verify physical applicability of the model is to evaluate the corresponding Boltzmann collision operator and to derive models for transport coefficients in terms of the collision kernel parameters. Comparison with experimental data for polytropic gases determines values of these parameters showing the physical relevance of the collision kernel.