Discrete velocity Boltzmann equation in the plane: stationary solutions

TL;DR

This paper establishes the existence of solutions to a specific class of discrete velocity Boltzmann equations in the plane, using advanced compactness techniques to handle the discrete velocities.

Contribution

It proves the existence of mild solutions for discrete velocity Boltzmann equations without colinear velocities, employing the Kolmogorov-Riesz theorem for compactness.

Findings

Existence of mild solutions proven

L1 compactness achieved via Kolmogorov-Riesz theorem

Applicable to equations with no colinear velocities

Abstract

The paper proves existence of mild solutions to normal discrete velocity Boltzmann equations sin the plane with no pair of colinear interacting velocities, and given ingoing boundary values. A key property is L1 compactness of the integrated collision frequency for a sequence of approximations. This is proved using the Kolmogorov-Riesz theorem, which here replaces the L1 compactness of velocity averages, not available when the velocities are discrete.