Discrete velocity Boltzmann equations in the plane: stationary solutions

Arkeryd, Leif; Nouri, Anne

doi:10.2140/apde.2023.16.1869

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Discrete velocity Boltzmann equations in the plane: stationary solutions

Leif Arkeryd and Anne Nouri

Vol. 16 (2023), No. 8, 1869–1884

DOI: 10.2140/apde.2023.16.1869

Abstract

We prove the 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. We remove an important restriction from a previous paper that all velocities point into the same half-space. 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.

Keywords

stationary Boltzmann equation, discrete coplanar velocities, normal model

Mathematical Subject Classification

Primary: 60K35, 82C40, 82C99

Milestones

Received: 28 May 2021

Revised: 29 November 2021

Accepted: 14 February 2022

Published: 16 October 2023

Authors

Leif Arkeryd
Mathematical Sciences Göteborg Sweden

Anne Nouri
Aix-Marseille University CNRS, I2M UMR 7373 Marseille France

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Vol. 16, No. 8, 2023