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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.18269 |
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| _version_ | 1866917416875851776 |
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| author | Sobah, Koudzo Togbévi Selom D'Almeida, Amah Séna |
| author_facet | Sobah, Koudzo Togbévi Selom D'Almeida, Amah Séna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18269 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 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 Sobah, Koudzo Togbévi Selom D'Almeida, Amah Séna Analysis of PDEs 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. |
| title | 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 |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.18269 |