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Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations

Xushan Huang, Moon-Jin Kang, Jeongho Kim, Hobin Lee

Abstract

We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in $H^1$ norm. Since our method works on the physical variables, we do not require that the anti-derivative variables belong to $L^2$ space as in \cite{HMS03,MM99}. Moreover, for the inflow case, we remove the assumption $γ\le 3$ in \cite{HMS03}. Our results are based on the method of $a$-contraction with shifts, as the first extension of the method to the boundary value problems.

Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations

Abstract

We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in norm. Since our method works on the physical variables, we do not require that the anti-derivative variables belong to space as in \cite{HMS03,MM99}. Moreover, for the inflow case, we remove the assumption in \cite{HMS03}. Our results are based on the method of -contraction with shifts, as the first extension of the method to the boundary value problems.
Paper Structure (31 sections, 21 theorems, 268 equations)

This paper contains 31 sections, 21 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Literature review
  3. Main results
  4. Main ideas for the proof
  5. Organization of the paper
  6. Preliminaries
  7. Estimates on the relative quantities
  8. Viscous shock wave
  9. Poincaré-type inequality
  10. A priori estimate and proof of the main theorem
  11. Local existence of solutions
  12. Construction of shift
  13. A priori estimates
  14. Proof of Theorem \ref{['thm:impermeable']} and Theorem \ref{['thm:inflow']}
  15. Relative Entropy estimates for the impermeable wall problem
  16. ...and 16 more sections

Key Result

Theorem 1.1

Assume $\gamma>1$. For a given constant state $(v_+,u_+)$ with $v_+>0$ and $u_+<0$, there exists $\delta_0,\varepsilon_0,\beta>0$ such that the following holds. For any $(v_-,u_-=0)$ satisfying RH_impermeable such that $|v_+-v_-|\sim |u_+-u_-|=|u_+|<\delta_0$, let $(\widetilde{v}(x-\sigma t),\wideti Then, the impermeable wall problem eq:NS--boundary_impermeable subject to the initial data $(v_0,u_

Theorems & Definitions (37)

  • Theorem 1.1: Impermeable wall problem
  • Theorem 1.2: Inflow problem
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • ...and 27 more