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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.02224 |
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| _version_ | 1866909674454908928 |
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| author | Eo, Saehoon Eun, Namhyun |
| author_facet | Eo, Saehoon Eun, Namhyun |
| contents | The Brenner-Navier-Stokes-Fourier (BNSF) system, introduced by Howard Brenner, was developed to address some deficiencies in the classical Navier-Stokes-Fourier system, based on the concept of volume velocity. We consider the one-dimensional BNSF system in Lagrangian mass coordinates, incorporating temperature-dependent transport coefficients, which yields a more physically realistic framework. We establish the existence and uniqueness of monotone traveling wave solutions (or viscous shocks) to the BNSF system with any positive $C^2$ dissipation coefficients, provided that the shock amplitude is sufficiently small. We utilize geometric singular perturbation theory as in the constant coefficient case [13]; however, due to the arbitrary nonlinearities of the coefficients, we employ the implicit function theorem, which grants robustness to our approach. This work is motivated by [12], which proves a contraction property of any large solutions to the BNSF system around the traveling wave solutions. Thus, we also derive some quantitative estimates on the traveling wave solutions that play a fundamental role in [12]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02224 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Traveling Wave Solutions to a Large Class of Brenner-Navier-Stokes-Fourier Systems Eo, Saehoon Eun, Namhyun Analysis of PDEs Mathematical Physics 76N15, 35Q30, 35C07, 35K65 The Brenner-Navier-Stokes-Fourier (BNSF) system, introduced by Howard Brenner, was developed to address some deficiencies in the classical Navier-Stokes-Fourier system, based on the concept of volume velocity. We consider the one-dimensional BNSF system in Lagrangian mass coordinates, incorporating temperature-dependent transport coefficients, which yields a more physically realistic framework. We establish the existence and uniqueness of monotone traveling wave solutions (or viscous shocks) to the BNSF system with any positive $C^2$ dissipation coefficients, provided that the shock amplitude is sufficiently small. We utilize geometric singular perturbation theory as in the constant coefficient case [13]; however, due to the arbitrary nonlinearities of the coefficients, we employ the implicit function theorem, which grants robustness to our approach. This work is motivated by [12], which proves a contraction property of any large solutions to the BNSF system around the traveling wave solutions. Thus, we also derive some quantitative estimates on the traveling wave solutions that play a fundamental role in [12]. |
| title | Traveling Wave Solutions to a Large Class of Brenner-Navier-Stokes-Fourier Systems |
| topic | Analysis of PDEs Mathematical Physics 76N15, 35Q30, 35C07, 35K65 |
| url | https://arxiv.org/abs/2507.02224 |