A consistent kinetic model for a two-component mixture with an application to plasma

Abstract

We consider a non reactive multi component gas mixture.We pro- pose a class of models, which can be easily generalized to multiple species. The two species mixture is modelled by a system of kinetic BGK equations featuring two interaction terms to account for momentum and energy transfer between the species.We prove consistency of our model: conservation proper- ties, positivity of the solutions for the space homogeneous case, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the space homogeneous case in the form of a global Maxwell distribution. Thus, we are able to derive the usual macroscopic conservation laws. In particular, by considering a mixture composed of ions and electrons, we derive the macroscopic equations of ideal MHD from our model

Hypocoercivity for a BGK model for gas mixtures

“…Moreover, in [21] consistency of this model, like conservation properties (conservation of the number of particles of each species, conservation of total momentum and conservation of total energy), positivity and the H-Theorem, is proven. Since the models from physicists mentioned above are special cases of the model proposed in [21], consistency of all these models is also proven. Another possible extension to gas mixtures was proposed by Andries, Aoki and Perthame in 2002 [3].…”

“…Just like the Boltzmann equation for gas mixtures that contains a sum of collision terms on the right-hand side, these kinds of models also have a sum of collision terms in the relaxation operator. In 2017, Klingenberg, Pirner and Puppo [21] proposed a kinetic model for gas mixtures which contains these often used models by physicists and engineers as special cases. Moreover, in [21] consistency of this model, like conservation properties (conservation of the number of particles of each species, conservation of total momentum and conservation of total energy), positivity and the H-Theorem, is proven.…”

“…Although large-time behavior of the monospecies BGK equations were intensively studied in the literature, but they are unknown for multispecies BGK systems. The purpose of this paper is to study the large-time behavior of a linearized version of the BGK model for gas mixtures presented in [21]. We study hypocoercivity of this linearized model in one-dimensional phase space and construct an entropy functional to prove exponential relaxation to equilibrium with explicit rates.…”

“…The outline of the paper is as follows: In subsection 2.1, we will present the BGK model for two species developed in [21]. In subsection 2.2, we perform a linearization of this model assuming that the distribution functions are close to equilibrium.…”

“…For the convenience of the reader, we want to briefly repeat it here. For more details, see [21]. We consider the position spacẽ T d := L 2Π T d , the d-dimensional torus of side length L. Then, we consider the following BGK…”