We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and Htheorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.

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The relaxation of rarefied gases of particles with the power-law interaction potentials Uϭ␣/r s , where 1 рsϽ4, is considered. The formation and evolution of the distribution function tails are investigated on the basis of the one-dimensional kinetic Landau-Fokker-Planck equation. For long times, the constructed asymptotic solutions have a propagating-wave appearance in the high velocity region. The analytical solutions are expressed explicitly in terms of the error function. The analytical consideration is accomplished by numerical calculations. The obtained analytical results are in a good agreement with the numerical simulation results.

We discuss some general properties of the Landau kinetic equation. In particular, the difference between the "true" Landau equation, which formally follows from classical mechanics, and the "generalized" Landau equation, which is just an interesting mathematical object, is stressed. We show how to approximate solutions to the Landau equation by the Wild sums. It is the so-called quasi-Maxwellian approximation related to Monte Carlo methods. This approximation can be also useful for mathematical problems. A model equation which can be reduced to a local nonlinear parabolic equation is also constructed in connection with existence of the strong solution to the initial value problem. A self-similar asymptotic solution to the Landau equation for large v and t is discussed in detail. The solution, earlier confirmed by numerical experiments, describes a formation of Maxwellian tails for a wide class of initial data concentrated in the thermal domain. It is shown that the corresponding rate of relaxation (fractional exponential function) is in exact agreement with recent mathematically rigorous estimates.

The relaxation process of a space uniform plasma composed of electrons and one species of ions is considered. For the varied initial electron and ion temperatures, the asymptotic behavior of the solutions, under m e /m i Ӷ1, are studied. Special attention has been paid to the deviation of relaxation from the classical picture, the latter being characterized by a weakly nonisothermic situation T e ӷ(m e /m i) 1/3 T i. An approach is developed for the detailed description of the relaxation. The perturbation of the electron distribution function, which has the character of a boundary layer for the cold electrons, is studied. The field of applicability of the well-known formulas for temperatures is extended and their corrections are obtained. The relaxation process of the two-temperature plasma is also considered numerically. A comparison of the results of the calculation with those of the asymptotic approach is made. The analytical results are confirmed by the numerical simulation results. ͓S1063-651X͑97͒02908-5͔