Asymptotic behavior toward viscous shocks for the outflow problem of barotropic Navier-Stokes equations
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| Format: | Preprint |
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2025
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| author | Kang, Moon-Jin Oh, HyeonSeop Wang, Yi |
| author_facet | Kang, Moon-Jin Oh, HyeonSeop Wang, Yi |
| contents | We study the large-time asymptotic stability of viscous shock profile to the outflow problem of barotropic Navier-Stokes equations on a half line. We consider the case when the far-field state as a right-end state of 2-Hugoniot shock curve belongs to the subsonic region or transonic curve. We employ the method of $a$-contraction with shifts, to prove that if the strength of viscous shock wave is small and sufficiently away from the boundary, and if a initial perturbation is small, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of time-dependent shift decays to zero as times goes to infinity, the shifted viscous shock still retains its original profile time-asymptotically.
Since the outflow problem in the Lagrangian mass coordinate leads to a free boundary value problem due to the absence of a boundary condition for the fluid density, we consider the problem in the Eulerian coordinate instead. Although the $a$-contraction method is technically more complicated in the Eulerian coordinate than in the Lagrangian one, this provides a more favorable framework by avoiding the difficulty arising from a free boundary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior toward viscous shocks for the outflow problem of barotropic Navier-Stokes equations Kang, Moon-Jin Oh, HyeonSeop Wang, Yi Analysis of PDEs We study the large-time asymptotic stability of viscous shock profile to the outflow problem of barotropic Navier-Stokes equations on a half line. We consider the case when the far-field state as a right-end state of 2-Hugoniot shock curve belongs to the subsonic region or transonic curve. We employ the method of $a$-contraction with shifts, to prove that if the strength of viscous shock wave is small and sufficiently away from the boundary, and if a initial perturbation is small, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of time-dependent shift decays to zero as times goes to infinity, the shifted viscous shock still retains its original profile time-asymptotically. Since the outflow problem in the Lagrangian mass coordinate leads to a free boundary value problem due to the absence of a boundary condition for the fluid density, we consider the problem in the Eulerian coordinate instead. Although the $a$-contraction method is technically more complicated in the Eulerian coordinate than in the Lagrangian one, this provides a more favorable framework by avoiding the difficulty arising from a free boundary. |
| title | Asymptotic behavior toward viscous shocks for the outflow problem of barotropic Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.08171 |