Long-time behavior towards viscous-dispersive shock for Navier-Stokes equations of Korteweg type
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913235018448896 |
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| author | Han, Sungho Kang, Moon-Jin Kim, Jeongho Lee, Hobin |
| author_facet | Han, Sungho Kang, Moon-Jin Kim, Jeongho Lee, Hobin |
| contents | We consider the so-called Naiver-Stokes-Korteweg(NSK) equations for the dynamics of compressible barotropic viscous fluids with internal capillarity. We handle the time-asymptotic stability in 1D of the viscous-dispersive shock wave that is a traveling wave solution to NSK as a viscous-dispersive counterpart of a Riemann shock. More precisely, we prove that when the prescribed far-field states of NSK are connected by a single Hugoniot curve, then solutions of NSK tend to the viscous-dispersive shock wave as time goes to infinity. To obtain the convergence, we extend the theory of $a$-contraction with shifts, used for the Navier-Stokes equations, to the NSK system. The main difficulty in analysis for NSK is due to the third-order derivative terms of the specific volume in the momentum equation. To resolve the problem, we introduce an auxiliary variable that is equivalent to the derivative of the specific volume. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time behavior towards viscous-dispersive shock for Navier-Stokes equations of Korteweg type Han, Sungho Kang, Moon-Jin Kim, Jeongho Lee, Hobin Analysis of PDEs We consider the so-called Naiver-Stokes-Korteweg(NSK) equations for the dynamics of compressible barotropic viscous fluids with internal capillarity. We handle the time-asymptotic stability in 1D of the viscous-dispersive shock wave that is a traveling wave solution to NSK as a viscous-dispersive counterpart of a Riemann shock. More precisely, we prove that when the prescribed far-field states of NSK are connected by a single Hugoniot curve, then solutions of NSK tend to the viscous-dispersive shock wave as time goes to infinity. To obtain the convergence, we extend the theory of $a$-contraction with shifts, used for the Navier-Stokes equations, to the NSK system. The main difficulty in analysis for NSK is due to the third-order derivative terms of the specific volume in the momentum equation. To resolve the problem, we introduce an auxiliary variable that is equivalent to the derivative of the specific volume. |
| title | Long-time behavior towards viscous-dispersive shock for Navier-Stokes equations of Korteweg type |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.09751 |