Stability of a Riemann Shock in a Physical Class: From Brenner-Navier-Stokes-Fourier to Euler
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908409695043584 |
|---|---|
| author | Eo, Saehoon Eun, Namhyun Kang, Moon-Jin |
| author_facet | Eo, Saehoon Eun, Namhyun Kang, Moon-Jin |
| contents | The stability of an irreversible singularity, such as a Riemann shock to the full Euler system, in the absence of any technical conditions on perturbations, remains a major open problem even within mono-dimensional framework. A natural approach to justify such stability is to consider vanishing dissipation (or viscosity) limits of physical viscous flows. We prove the existence of vanishing dissipation limits, on which a Riemann shock of small amplitude is stable (up to a time-dependent shift) and unique. Thus, a Riemann weak shock is rigid (not turbulent) under physical disturbances. We adopt the Brenner-Navier-Stokes-Fourier system, based on the bi-velocity theory, as a physical viscous model. The key ingredient of the proof is the uniform stability of the viscous shock with respect to the viscosity strength. The uniformity is ensured by contraction estimates of any large perturbations around the shock. The absence of any restrictions on size of initial perturbations forces us to handle extreme values of density and temperature, which constitutes the most challenging part of our analysis. We use the method of a-contraction with shifts, but we improve it by introducing a more delicate analysis of the localizing effect given by viscous shock derivatives. This improvement possesses a degree of robustness that renders it applicable to a wide range of models. This is the first resolution for the challenging open problem on the "unconditional" stability and uniqueness of Riemann shock solutions to the full Euler system in a class of vanishing physical dissipation limits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03613 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of a Riemann Shock in a Physical Class: From Brenner-Navier-Stokes-Fourier to Euler Eo, Saehoon Eun, Namhyun Kang, Moon-Jin Analysis of PDEs Mathematical Physics 76N15, 35B35, 35Q30 The stability of an irreversible singularity, such as a Riemann shock to the full Euler system, in the absence of any technical conditions on perturbations, remains a major open problem even within mono-dimensional framework. A natural approach to justify such stability is to consider vanishing dissipation (or viscosity) limits of physical viscous flows. We prove the existence of vanishing dissipation limits, on which a Riemann shock of small amplitude is stable (up to a time-dependent shift) and unique. Thus, a Riemann weak shock is rigid (not turbulent) under physical disturbances. We adopt the Brenner-Navier-Stokes-Fourier system, based on the bi-velocity theory, as a physical viscous model. The key ingredient of the proof is the uniform stability of the viscous shock with respect to the viscosity strength. The uniformity is ensured by contraction estimates of any large perturbations around the shock. The absence of any restrictions on size of initial perturbations forces us to handle extreme values of density and temperature, which constitutes the most challenging part of our analysis. We use the method of a-contraction with shifts, but we improve it by introducing a more delicate analysis of the localizing effect given by viscous shock derivatives. This improvement possesses a degree of robustness that renders it applicable to a wide range of models. This is the first resolution for the challenging open problem on the "unconditional" stability and uniqueness of Riemann shock solutions to the full Euler system in a class of vanishing physical dissipation limits. |
| title | Stability of a Riemann Shock in a Physical Class: From Brenner-Navier-Stokes-Fourier to Euler |
| topic | Analysis of PDEs Mathematical Physics 76N15, 35B35, 35Q30 |
| url | https://arxiv.org/abs/2411.03613 |