On the existence and uniqueness of classical solution for an initial-boundary value problem for a discrete Boltzmann system in two space dimensions

TL;DR

This paper proves the existence and uniqueness of a classical positive solution for a two-dimensional discrete Boltzmann system with boundary conditions, providing bounds for the solution and its derivatives.

Contribution

It establishes the first rigorous proof of existence and uniqueness for this class of discrete Boltzmann systems in two dimensions.

Findings

Existence of a unique classical positive solution

Boundedness of the solution and its derivatives

Application of fixed point methods

Abstract

The initial-boundary value problem for the two-dimensional regular four-velocity discrete Boltzmann system is analyzed in a rectangle. The existence and uniqueness of a classical global positive solution, bounded with its first partial derivatives are proved for a range of bounded data by the use of fixed points tools. A bound for the solution and its partial derivatives is provided.