Existence, Uniqueness and Positivity of solutions for BGK models for mixtures

TL;DR

This paper investigates mathematical properties such as existence, uniqueness, and positivity of solutions for two types of BGK kinetic models describing multi-component gas mixtures, and applies these results to energy exchange in macroscopic equations.

Contribution

It provides the first rigorous proofs of existence, uniqueness, and positivity for both types of BGK models for gas mixtures, and applies the first model to energy exchange problems.

Findings

Proved existence, uniqueness, and positivity for models with multiple collision terms.

Established similar properties for models with a single collision term.

Applied the first model to determine energy exchange functions in macroscopic equations.

Abstract

We consider kinetic models for a multi component gas mixture without chemical reactions. In the literature, one can find two types of BGK models in order to describe gas mixtures. One type has a sum of BGK type interaction terms in the relaxation operator, for example the model described by Klingenberg, Pirner and Puppo, 2017 which contains well-known models of physicists and engineers for example Hamel, 1965, and Gross and Krook, 1956, as special cases. The other type contains only one collision term on the right-hand side, for example the well-known model of Andries, Aoki and Perthame, 2002. For each of these two models we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie.