Amah Séna D'Almeida | Searcharxiv

arXiv2026

Global existence and uniqueness for the planar four-velocity Broadwell model with arbitrary $C^{1}$ bounded initial and boundary data in a rectangular domain

This paper establishes the global existence of classical solutions to the initial-boundary value problem for the planar four-velocity Broadwell model in a multidimensional setting. While uniqueness was already established in our previous work (which addressed existence under the assumption of sufficiently small, bounded, and $C^{1}$-regular initial and boundary data), the objective of the present study is to completely lift this size restriction. By extending the previously developed fixed-point framework, we prove that the methodology remains highly effective for arbitrary initial and boundary data. Within this generalized setting, both the solutions and their first-order partial derivatives are shown to be well-defined and to exist globally for all times.

math.AP↗

arXiv2026

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

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.

math.AP↗