Existence, uniqueness and positivity of solutions for BGK models for mixtures

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 [20] which contains wellknown models of physicists and engineers for example Hamel [16] and Gross and Krook [15] 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 [1]. For each of these two models [20] and [1], we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model [20] in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie [11].

BGK model of the multi-species Uehling-Uhlenbeck equation

“…Examples include the model of Gross and Krook [29], the model of Hamel [32], the model of Greene [27], the model of Garzo, Santos and Brey [26], the model of Sofonea and Sekerka [57], the model by Andries, Aoki and Perthame [1], the model of Brull, Pavan and Schneider [13], the model of Klingenberg, Pirner and Puppo [40], the model of Haack, Hauck, Murillo [31] and the model of Bobylev, Bisi, Groppi, Spiga [10]. BGK models have also been extended to ES-BGK models, polyatomic molecules or chemically reactive gas mixtures; see for example [8,9,14,28,38,39,41,49,60]. BGK models are often used in applications because they give rise to efficient numerical computations as compared to models with Boltzmann collision terms [5,6,16,17,24,48,53,54].…”

Global existence of weak solutions for Navier–Stokes-BGK system

“…Perthame and Pulvirenti later studied the existence of unique mild solution in a weighted L ∞ space [31]. These works were fruitfully extended to several directions: gases in the presence of external forces [43], plasma [50,51], solutions in L p spaces [49], ellipsoidal extension [47] and gas mixture problems [23]. The existence of classical solutions and its exponential stabilization near equilibrium are studied in [46,48].…”

Numerical evaluation of novel kinetic models for binary gas mixture flows

“…[9][10][11][12] and multirelaxation models, e.g. [13][14][15] . A new consistent BGK mixture model was introduced recently 16 for both Maxwellian and arbitrary potential molecules.…”