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Asymptotic behavior toward viscous shocks for the outflow problem of barotropic Navier-Stokes equations

Moon-Jin Kang, HyeonSeop Oh, Yi Wang

Abstract

We study the large-time asymptotic stability of viscous shock profile to the outflow problem of barotropic Navier-Stokes equations on a half line. We consider the case when the far-field state as a right-end state of 2-Hugoniot shock curve belongs to the subsonic region or transonic curve. We employ the method of $a$-contraction with shifts, to prove that if the strength of viscous shock wave is small and sufficiently away from the boundary, and if a initial perturbation is small, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of time-dependent shift decays to zero as times goes to infinity, the shifted viscous shock still retains its original profile time-asymptotically. Since the outflow problem in the Lagrangian mass coordinate leads to a free boundary value problem due to the absence of a boundary condition for the fluid density, we consider the problem in the Eulerian coordinate instead. Although the $a$-contraction method is technically more complicated in the Eulerian coordinate than in the Lagrangian one, this provides a more favorable framework by avoiding the difficulty arising from a free boundary.

Asymptotic behavior toward viscous shocks for the outflow problem of barotropic Navier-Stokes equations

Abstract

We study the large-time asymptotic stability of viscous shock profile to the outflow problem of barotropic Navier-Stokes equations on a half line. We consider the case when the far-field state as a right-end state of 2-Hugoniot shock curve belongs to the subsonic region or transonic curve. We employ the method of -contraction with shifts, to prove that if the strength of viscous shock wave is small and sufficiently away from the boundary, and if a initial perturbation is small, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of time-dependent shift decays to zero as times goes to infinity, the shifted viscous shock still retains its original profile time-asymptotically. Since the outflow problem in the Lagrangian mass coordinate leads to a free boundary value problem due to the absence of a boundary condition for the fluid density, we consider the problem in the Eulerian coordinate instead. Although the -contraction method is technically more complicated in the Eulerian coordinate than in the Lagrangian one, this provides a more favorable framework by avoiding the difficulty arising from a free boundary.
Paper Structure (21 sections, 14 theorems, 193 equations)

This paper contains 21 sections, 14 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Viscous shock wave
  5. Poincaré type inequality
  6. A priori estimate and proof of Theorem \ref{['thm:main']}
  7. Local existence of solutions
  8. Construction of weight and shift
  9. A priori estimates
  10. Proof of Theorem \ref{['thm:main']}
  11. $L^2$ energy estimates
  12. Relative entropy method
  13. Maximization in terms of $u-{\widetilde{u}}$
  14. Estimates for $\dot{X}Y + B - G$
  15. Estimate on the boundary terms
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume $\gamma > 1$. For a given constant $(\rho_+, u_+)$ satisfying $u_+ <0$ and cond:U+, there exist constants $\delta_0, \varepsilon_0$ such that the following holds. For any $u_- < 0$ satisfying $u_->u_+$ and $|u_- - u_+| < \delta_0$, let $\rho_->0$ be the (unique) constant state such that $(\rh Then, the outflow problem NS - BC admits an unique global-in-time solution $(\rho, u)(t,x)$ satisfy

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Remark 4.1
  • ...and 14 more