Global-in-time existence and uniqueness of classical solutions to the unsteady initial-boundary value problem for the four-velocity planar Broadwell model in a rectangular domain

TL;DR

This paper proves the global existence and uniqueness of classical solutions for the unsteady four-velocity Broadwell model in a rectangular domain, advancing the mathematical understanding of discrete velocity models in gas dynamics.

Contribution

It establishes the first rigorous proof of global-in-time classical solutions for the nonstationary four-velocity Broadwell system, using fixed point and a priori estimate techniques.

Findings

Proves global-in-time existence of classical solutions.

Shows solutions and derivatives are uniformly bounded.

Provides a well-posedness framework for the Broadwell model.

Abstract

Since the pioneering work of James E. Broadwell, discrete velocity models (DVMs) have played a fundamental role in approximating the Boltzmann equation and in the analysis of non-equilibrium gas dynamics. Despite their apparent simplicity, many fundamental analytical questions remain open, in particular the global existence and uniqueness of classical solutions, even for the widely studied four-velocity Broadwell model. In this paper, we establish the global-in-time existence and uniqueness of classical solutions to the nonstationary four-velocity Broadwell system in a rectangular domain. The analysis is carried out in a class of continuous functions possessing, except possibly on a finite number of planes, continuous first-order partial derivatives. Our approach is based on fixed point arguments combined with suitable a priori estimates that provide uniform bounds on the solution and its first-order partial derivatives. These bounds ensure that the solution remains controlled for all time and can be extended globally. We prove the existence of a unique bounded continuous solution whose first-order partial derivatives are also bounded. These results provide a rigorous well-posedness framework for this prototypical discrete velocity model and contribute to a deeper understanding of the analytical properties of discrete velocity models, which serve as systematic approximations of the Boltzmann equation in the study of non-equilibrium gas dynamics.