Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910773912010752 |
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| author | Huang, Xushan Kang, Moon-Jin Kim, Jeongho Lee, Hobin |
| author_facet | Huang, Xushan Kang, Moon-Jin Kim, Jeongho Lee, Hobin |
| contents | We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in $H^1$ norm. Since our method works on the physical variables, we do not require that the anti-derivative variables belong to $L^2$ space as in \cite{HMS03,MM99}. Moreover, for the inflow case, we remove the assumption $γ\le 3$ in \cite{HMS03}. Our results are based on the method of $a$-contraction with shifts, as the first extension of the method to the boundary value problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations Huang, Xushan Kang, Moon-Jin Kim, Jeongho Lee, Hobin Analysis of PDEs We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in $H^1$ norm. Since our method works on the physical variables, we do not require that the anti-derivative variables belong to $L^2$ space as in \cite{HMS03,MM99}. Moreover, for the inflow case, we remove the assumption $γ\le 3$ in \cite{HMS03}. Our results are based on the method of $a$-contraction with shifts, as the first extension of the method to the boundary value problems. |
| title | Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.03214 |