Abstract
We consider the Bathnagar–Gross–Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications, one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom such as rotations or vibrations (polyatomic molecules). Therefore, here, we will present recent BGK models for gas mixtures for mono- and polyatomic particles and the existing mathematical theory for these models.
Highlights
In this paper, we concern ourselves with a kinetic description of gas mixtures
If we are close to equilibrium [5,6], the complicated interaction terms of the Boltzmann equation can be simplified by a BGK approximation
The outline of the paper is as follows: In Section 2, we will present typical ansatzes for modelling gas mixtures with the BGK model and a review on recent results concerning the existence of solutions and large-time behaviour
Summary
We concern ourselves with a kinetic description of gas mixtures. In the case of mono atomic molecules and two species, this is usually done with the Boltzmann equation for the distribution functions f 1 = f 1 ( x, v, t), f 2 = f 2 ( x, v, t), see for example [1,2]. If we are close to equilibrium [5,6], the complicated interaction terms of the Boltzmann equation can be simplified by a BGK approximation This consists of a collision frequency νij n j multiplied by the deviation of the distribution functions from a local Maxwell distribution. This BGK model with velocity-dependent collision frequency should be constructed in a way such that it satisfies the conservation properties That this works one has to replace the Maxwell distribution by a different function, for details see [7]. The outline of the paper is as follows: In Section 2, we will present typical ansatzes for modelling gas mixtures with the BGK model and a review on recent results concerning the existence of solutions and large-time behaviour.
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