Discrete velocity Boltzmann equation in the plane: stationary solutions

Abstract

The paper proves existence of stationary mild solutions for normal discrete velocity Boltzmann equations in the plane with no pair of colinear interacting velocities and given ingoing boundary values. An important restriction of all velocities pointing into the same half-space in a previous paper is removed in this paper. A key property is L 1 compactness of integrated collision frequency for a sequence of approximations. This is proven using the Kolmogorov-Riesz theorem, which here replaces the L 1 compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete. .

Stationary solutions to the Boltzmann equation in the plane

“…This is inspired by recent results for discrete velocity models for the Boltzmann equation where averaging lemmas do not hold and new arguments are required. In [7], [8], [9] a weaker property than L 1 compactness of averages in velocity, i.e. the L 1 compactness of the integrated collision frequencies of a sequence of approximations is proven.…”

“…It strongly depends on the two-dimensional spatial dimension. In this paper we use the tools developped in [7], [8], [9] and do not use any averaging lemma. Work is in progress to fill a gap in the proof of Lemma 4.1 of [10] that uses these techniques in the discrete velocity evolutionary case.…”