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arXiv · 2402.09751

Long-time behavior towards viscous-dispersive shock for Navier-Stokes equations of Korteweg type

Sungho Han·Moon-Jin Kang·Jeongho Kim·Hobin Lee
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Abstract

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.

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Sungho Han, Moon-Jin Kang, Jeongho Kim, Hobin Lee. 2024-02-15. Long-time behavior towards viscous-dispersive shock for Navier-Stokes equations of Korteweg type. https://arxiv.org/abs/2402.09751

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