Group-invariant solutions of kinetic equations (Q792170)

Similarity solutions of the Bhatnagar-Gross-Krook equation of kinetic theory are obtained by group theoretic methods. Solutions f depending on space x, time t, and velocity v are sought by rewriting the equations as an infinite system of partial differential equation in x and t for the moments \(f_ k\) of f with respect to v. This system is reduced to an infinite system of ordinary differential equations for the similarity solutions. By the group theoretic method various possible similarity forms are found and in several special cases the full distribution function or some of its moments are constructed explicitly. The generalization of this method to the B-G-K equation with a variable cross section and to the full Boltzmann equation are outlined.