Time-asymptotic stability of generic Riemann solutions for Boltzmann equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916556193136640 |
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| author | Wang, Yi Yu, Qiuyang |
| author_facet | Wang, Yi Yu, Qiuyang |
| contents | Time-asymptotic stability of generic Riemann solution, consisting of a rarefaction wave, a contact discontinuity and a shock, for the one-dimensional Boltzmann equation, has been a long-standing open problem in kinetic theory. In this paper, we proved that the composite waves of generic Riemann profile including the inviscid self-similar rarefaction wave, the viscous contact wave (i.e., the viscous version of contact discontinuity) and the viscous shock profile with the time-dependent shift to both macroscopic and microscopic components are nonlinearly stable for the one-dimensional Boltzmann equation, by the first using the $a$-contraction method to the Boltzmann equation. Compared with the compressible Navier-Stokes-Fourier equations, the new difficulties here lie in the microscopic effects of the Boltzmann shock profile and their interactions and/or couplings with the rarefaction wave, viscous contact wave and the macroscopic components from the macro-micro decomposition of the Boltzmann equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_04399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time-asymptotic stability of generic Riemann solutions for Boltzmann equation Wang, Yi Yu, Qiuyang Analysis of PDEs Mathematical Physics Time-asymptotic stability of generic Riemann solution, consisting of a rarefaction wave, a contact discontinuity and a shock, for the one-dimensional Boltzmann equation, has been a long-standing open problem in kinetic theory. In this paper, we proved that the composite waves of generic Riemann profile including the inviscid self-similar rarefaction wave, the viscous contact wave (i.e., the viscous version of contact discontinuity) and the viscous shock profile with the time-dependent shift to both macroscopic and microscopic components are nonlinearly stable for the one-dimensional Boltzmann equation, by the first using the $a$-contraction method to the Boltzmann equation. Compared with the compressible Navier-Stokes-Fourier equations, the new difficulties here lie in the microscopic effects of the Boltzmann shock profile and their interactions and/or couplings with the rarefaction wave, viscous contact wave and the macroscopic components from the macro-micro decomposition of the Boltzmann equation. |
| title | Time-asymptotic stability of generic Riemann solutions for Boltzmann equation |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2501.04399 |