Vanishing dissipation limit for non-isentropic Navier-Stokes equations with shock data
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913367452549120 |
|---|---|
| author | Huang, Feimin Wang, Teng |
| author_facet | Huang, Feimin Wang, Teng |
| contents | This paper is concerned with the vanishing dissipation limiting problem of one-dimensional non-isentropic Navier-Stokes equations with shock data. The limiting problem was solved in 1989 by Hoff-Liu in [13] for isentropic gas with single shock, but was left open for non-isentropic case. In this paper, we solve the non-isentropic case, i.e., we first establish the global existence of solutions to the non-isentropic Navier-Stokes equations with initial discontinuous shock data, and then show these solutions converge in $L^{\infty}$ norm to a single shock wave of the corresponding Euler equations away from the shock curve in any finite time interval, as both the viscosity and heat-conductivity tend to zero.
Different from [13] in which an integrated system was essentially used, motivated by [21,22], we introduce a time-dependent shift $\mathbf{X}^\varepsilon(t)$ to the viscous shock so that a weighted Poincaré inequality can be applied to overcome the difficulty generated from the ``bad" sign of the derivative of viscous shock velocity, and the anti-derivative technique is not needed. We also obtain an intrinsic property of non-isentropic viscous shock, see Lemma 2.2 below. With the help of Lemma 2.2, we can derive the desired uniform a priori estimates of solutions, which can be shown to converge in $L^{\infty}$ norm to a single inviscid shock in any given finite time interval away from the shock, as the vanishing dissipation limit. Moreover, the shift $\mathbf{X}^\varepsilon(t)$ tends to zero in any finite time as viscosity tends to zero. The proof consists of a scaling argument, $L^2$-contraction technique with time-dependent shift to the shock, and relative entropy method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02067 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Vanishing dissipation limit for non-isentropic Navier-Stokes equations with shock data Huang, Feimin Wang, Teng Analysis of PDEs This paper is concerned with the vanishing dissipation limiting problem of one-dimensional non-isentropic Navier-Stokes equations with shock data. The limiting problem was solved in 1989 by Hoff-Liu in [13] for isentropic gas with single shock, but was left open for non-isentropic case. In this paper, we solve the non-isentropic case, i.e., we first establish the global existence of solutions to the non-isentropic Navier-Stokes equations with initial discontinuous shock data, and then show these solutions converge in $L^{\infty}$ norm to a single shock wave of the corresponding Euler equations away from the shock curve in any finite time interval, as both the viscosity and heat-conductivity tend to zero. Different from [13] in which an integrated system was essentially used, motivated by [21,22], we introduce a time-dependent shift $\mathbf{X}^\varepsilon(t)$ to the viscous shock so that a weighted Poincaré inequality can be applied to overcome the difficulty generated from the ``bad" sign of the derivative of viscous shock velocity, and the anti-derivative technique is not needed. We also obtain an intrinsic property of non-isentropic viscous shock, see Lemma 2.2 below. With the help of Lemma 2.2, we can derive the desired uniform a priori estimates of solutions, which can be shown to converge in $L^{\infty}$ norm to a single inviscid shock in any given finite time interval away from the shock, as the vanishing dissipation limit. Moreover, the shift $\mathbf{X}^\varepsilon(t)$ tends to zero in any finite time as viscosity tends to zero. The proof consists of a scaling argument, $L^2$-contraction technique with time-dependent shift to the shock, and relative entropy method. |
| title | Vanishing dissipation limit for non-isentropic Navier-Stokes equations with shock data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2306.02067 |