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The method of $a$-contraction with shifts used for long-time behavior toward viscous shock

Sungho Han, Moon-Jin Kang, Hobin Lee

Abstract

We revisit the method of $a$-contraction with shifts used for long-time behavior of barotropic Navier-Stokes flows perturbed from a Riemann shock. For the usage of the method of $a$-contraction with shifts, we do not employ the effective velocity $h$ variable even for higher order estimates. This approach would be important when handling the barotropic Navier-Stokes system with other effects, for example, such as capillary effect and boundary effect.

The method of $a$-contraction with shifts used for long-time behavior toward viscous shock

Abstract

We revisit the method of -contraction with shifts used for long-time behavior of barotropic Navier-Stokes flows perturbed from a Riemann shock. For the usage of the method of -contraction with shifts, we do not employ the effective velocity variable even for higher order estimates. This approach would be important when handling the barotropic Navier-Stokes system with other effects, for example, such as capillary effect and boundary effect.

Paper Structure

This paper contains 21 sections, 14 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimates on the relative quantities
  4. Viscous shock wave
  5. Useful $O(1)$-constants
  6. A priori estimate and Proof of Theorem \ref{['thm:main']}
  7. Local existence
  8. Construction of weight function
  9. Construction of shift
  10. A priori estimate
  11. Proof of Theorem \ref{['thm:main']}
  12. Estimate on the weighted relative entropy with the shift
  13. Relative entropy method
  14. Maximization on $p(v)-p(\widetilde{v})$
  15. Estimate of $\mathcal{R}_1$
  16. ...and 6 more sections

Key Result

Theorem 1.1

For a given state $(v_+,u_+)\in\mathbb R^+\times\mathbb R$, there exist positive constants $\delta_0$, and $\varepsilon_0$ such that the following holds: For any $(v_-,u_-)$ on the 2-shock curve $S_2(v_+,u_+)$, satisfying the Rankine-Hugoniot condition RH with $|v_+-v_-|<\delta_0$, consider the 2-vi where $\mathbb{R}_-:=-\mathbb{R}_+=(-\infty,0)$. Then, the Navier-Stokes system eq:NS admits a uniq

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • ...and 13 more