Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

A consistent kinetic model for a two-component mixture of polyatomic molecules

Pages: 149 – 173



Christian Klingenberg (Department of Mathematics, Würzburg University, Würzburg, Germany)

Marlies Pirner (Department of Mathematics, Würzburg University, Würzburg, Germany)

Gabriella Puppo (Department of Science and High Technology, Universitá degli studi dell’ Insubria, Como, Italy)


We consider a multi component gas mixture with translational and internal energy degrees of freedom assuming that the number of particles of each species remains constant. We will illustrate the derived model in the case of two species, but the model can be easily generalized to multiple species. The two species are allowed to have different degrees of freedom in internal energy and are modelled by a system of kinetic ES-BGK equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the temperature, H-theorem and convergence to a global equilibrium in the form of a global Maxwell distribution. Thus, we are able to derive the usual macroscopic conservation laws. For numerical purposes we apply the Chu reduction to the developed model for polyatomic gases and give an application for a gas consisting of a monoatomic and a diatomic species.


multi-fluid mixture, kinetic model, ES-BGK approximation, polyatomic molecules

2010 Mathematics Subject Classification

35Q20, 35Q79, 65Z05, 82C40

Received 3 August 2017

Accepted 3 November 2018

Published 30 May 2019