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Contraction of viscous-dispersive shocks: Zero viscosity-capillarity limits

Namhyun Eun, Moon-Jin Kang, Jeongho Kim

Abstract

We prove the contraction property of any large solution perturbed from a viscous-dispersive shock wave of the Navier--Stokes--Korteweg (NSK) system. The contraction holds up to a dynamical shift, since the contraction is measured by the relative entropy that is locally $L^2$. We use the contraction property to show the global existence of large solution perturbed from a viscous-dispersive shock wave. To prove the contraction property, we first employ the effective velocity to transform the NSK system into the system of two degenerate parabolic equations, then apply the method of $a$-contraction with shifts. The contraction property does not depend on the strengths of viscosity and capillarity. Based on this uniformity, we show the existence of zero viscosity-capillarity limits of solutions to the NSK system, on which Riemann shocks are unique and stable up to shifts.

Contraction of viscous-dispersive shocks: Zero viscosity-capillarity limits

Abstract

We prove the contraction property of any large solution perturbed from a viscous-dispersive shock wave of the Navier--Stokes--Korteweg (NSK) system. The contraction holds up to a dynamical shift, since the contraction is measured by the relative entropy that is locally . We use the contraction property to show the global existence of large solution perturbed from a viscous-dispersive shock wave. To prove the contraction property, we first employ the effective velocity to transform the NSK system into the system of two degenerate parabolic equations, then apply the method of -contraction with shifts. The contraction property does not depend on the strengths of viscosity and capillarity. Based on this uniformity, we show the existence of zero viscosity-capillarity limits of solutions to the NSK system, on which Riemann shocks are unique and stable up to shifts.
Paper Structure (25 sections, 26 theorems, 252 equations)

This paper contains 25 sections, 26 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Transformation of the system
  4. Proof of Theorem \ref{['thm_main3']}
  5. Preliminaries for the method of $a$-contraction with shifts
  6. Global and local estimates on the relative quantities
  7. Properties of small shock waves
  8. Relative entropy method
  9. Construction of the weight function and shift
  10. Maximization in terms of $h-\tilde{h}$
  11. Main proposition
  12. Proof of Proposition \ref{['prop:main']}
  13. Sharp estimate near a shock wave
  14. Truncation of large values of $\left|p(v)-p(\tilde{v})\right|$
  15. Estimates on the parabolic bad terms
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\gamma,\alpha$ be any constants satisfying Assume $c\in[0,\frac{9}{100}]$ for the constant in kappa. Then, for a given constant state $(v_-,u_-) \in {\mathbb R}^+\times{\mathbb R}$, there exist constants $\varepsilon_0,\delta_0>0$ such that the following holds. Fix any $\varepsilon\in(0,\varepsilon_0),$$\lambda\in(\delta_0^{-1}\varepsilon,\d and

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1: EEKO-ISOKV-JEMS21
  • proof
  • ...and 40 more