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

Sungho Han, Moon-Jin Kang, Jeongho Kim, Hobin Lee

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.

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

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 -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.
Paper Structure (23 sections, 16 theorems, 321 equations, 1 figure)

This paper contains 23 sections, 16 theorems, 321 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimates on the relative quantities
  4. Viscous-dispersive shock wave
  5. Useful $O(1)$-constants
  6. Augmented system
  7. A priori estimate and Proof of Theorem \ref{['thm:main']}
  8. Local existence
  9. Construction of shift
  10. A priori estimate
  11. Global existence of perturbed solution
  12. Time-asymptotic behavior
  13. Estimate on the weighted relative entropy with the shift
  14. Construction of weight function
  15. Relative entropy method
  16. ...and 8 more sections

Key Result

Theorem 1.1

For a given state $(v_+,u_+)\in\mathbb R^+\times\mathbb R$, there exist positive constants $C_0,\delta_0$, and $\varepsilon_1$ such that the following holds. For any $(v_-,u_-)$ on the 2-shock curve $S_2(v_+,u_+)$, that is, satisfying the Rankine-Hugoniot condition RH, such that $|v_+-v_-|<\delta_0$ where $\mathbb{R}_-:=-\mathbb{R}_+=(-\infty,0)$. Then, the Navier-Stokes-Korteweg system eq:NSK adm

Figures (1)

  • Figure 1: Profiles of $\widetilde{v}$ for weak shock $\delta_S=0.05$ (left) and for strong shock $\delta_S=0.5$ (right), when the right-end state is fixed as $v_+=0.7$. The profile of the small shock is monotone, while that of the large shock has an oscillation. We also note that, even in the small shock case, the viscous-dispersive shock is not symmetric with respect to the inflection point, unlike the viscous shock of the classical Navier-Stokes equations.

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Proposition 3.1
  • proof
  • ...and 21 more