Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917304276615168 |
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| author | Huang, Xushan Lee, Hobin Oh, HyeonSeop |
| author_facet | Huang, Xushan Lee, Hobin Oh, HyeonSeop |
| contents | We investigate the time-asymptotic stability of solutions to the one-dimensional Navier-Stokes-Fourier system in the half-space, focusing on the outflow and impermeable wall problems. When the prescribed boundary and far-field conditions form an outgoing viscous shock, we prove that the solution converges to the viscous shock profile, up to a dynamical shift, provided that the initial perturbation and the shock amplitude are sufficiently small. In order to obtain our results, we employ the method of $a$-contraction with shifts. Although the impermeable wall problem is technically simpler to analyze in Lagrangian mass coordinates, the outflow problem leads to a free boundary in that framework. Therefore, we use Eulerian coordinates to provide a unified approach to both problems. This is the first result on the time-asymptotic stability of viscous shocks for initial-boundary value problems of the Navier-Stokes-Fourier system for the outflow and impermeable wall cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02215 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems Huang, Xushan Lee, Hobin Oh, HyeonSeop Analysis of PDEs We investigate the time-asymptotic stability of solutions to the one-dimensional Navier-Stokes-Fourier system in the half-space, focusing on the outflow and impermeable wall problems. When the prescribed boundary and far-field conditions form an outgoing viscous shock, we prove that the solution converges to the viscous shock profile, up to a dynamical shift, provided that the initial perturbation and the shock amplitude are sufficiently small. In order to obtain our results, we employ the method of $a$-contraction with shifts. Although the impermeable wall problem is technically simpler to analyze in Lagrangian mass coordinates, the outflow problem leads to a free boundary in that framework. Therefore, we use Eulerian coordinates to provide a unified approach to both problems. This is the first result on the time-asymptotic stability of viscous shocks for initial-boundary value problems of the Navier-Stokes-Fourier system for the outflow and impermeable wall cases. |
| title | Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.02215 |