Exact solutions of the Boltzmann—Maxwell kinetic equation

Abstract

Exact solutions of the following topics are reviewed: locally Maxwell solutions, spatially uniform relaxation of a binary mixture of gases (moment solutions and a generalization of the Bobylev—Krook—Wu solutions), homoenergetic affine flows (the Galkin—Truesdell class of exact solutions), spherical outflow-inflow (the Nikol'skii transformation), dominant solutions and power solutions. New results are obtained on the second and third topics. Particular attention is devoted to the qualitative properties of the solutions, which are of fairly general interest. We mean by an exact solution a solution in explicit form, i.e. in terms of elementary or transcendental functions, of the non-linear Boltzmann—Maxwell kinetic equation (the distribution function) or the Maxwell transport equations (the moments of the distribution functions). The latter are also called moment solutions of the kinetic equation. The molecules of the gas are mainly assumed to be Maxwellian, when the coefficient of viscosity and thermal conductivity depend linearly on the temperature.