Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence to superposition of boundary layer, rarefaction and shock for the 1D Navier-Stokes equations

Sungho Han, Moon-Jin Kang, Jeongho Kim, Nayeon Kim, HyeonSeop Oh

TL;DR

This work resolves the long-standing stability problem for the subsonic inflow of the 1D barotropic Navier–Stokes equations by proving the asymptotic stability of the complex boundary-layer–rarefaction–shock superposition. The authors adapt the a-contraction with shifts framework to a weighted relative-entropy setting, introduce a dynamical shift X(t) for the viscous shock, and construct a smooth approximate rarefaction to localize the leading-order dynamics near the shock. They derive comprehensive L^2 and H^1 a priori estimates, control wave interactions, and show global existence plus convergence of the solution to the BL+1-rarefaction+2-shock profile up to a vanishing shift rate. This completes the stability classification for subsonic inflow boundary values and provides a rigorous mechanism for the time-asymptotic stabilization of complex wave interactions in 1D NS flows.

Abstract

We establish the asymptotic stability of solutions to the inflow problem for the one-dimensional barotropic Navier-Stokes equations in half space. When the boundary value is located at the subsonic regime, all the possible thirteen asymptotic patterns are classified in \cite{M01}. We consider the most complicated pattern, the superposition of the boundary layer solution, the 1-rarefaction wave, and the viscous 2-shock waves. In this superposition, the boundary layer is degenerate and large. We prove that, if the strengths of the rarefaction wave and shock wave are small, and if the initial data is a small perturbation of the superposition, then the solution asymptotically converges to the superposition up to a dynamical shift for the shock. As a corollary, our result implies the asymptotic stability for the simpler case where the superposition consists of the degenerate boundary layer solution and the viscous 2-shock. Therefore, we complete the study of the asymptotic stability of the inflow problem for the 1D barotropic Navier-Stokes equations for subsonic boundary values.

Convergence to superposition of boundary layer, rarefaction and shock for the 1D Navier-Stokes equations

TL;DR

This work resolves the long-standing stability problem for the subsonic inflow of the 1D barotropic Navier–Stokes equations by proving the asymptotic stability of the complex boundary-layer–rarefaction–shock superposition. The authors adapt the a-contraction with shifts framework to a weighted relative-entropy setting, introduce a dynamical shift X(t) for the viscous shock, and construct a smooth approximate rarefaction to localize the leading-order dynamics near the shock. They derive comprehensive L^2 and H^1 a priori estimates, control wave interactions, and show global existence plus convergence of the solution to the BL+1-rarefaction+2-shock profile up to a vanishing shift rate. This completes the stability classification for subsonic inflow boundary values and provides a rigorous mechanism for the time-asymptotic stabilization of complex wave interactions in 1D NS flows.

Abstract

We establish the asymptotic stability of solutions to the inflow problem for the one-dimensional barotropic Navier-Stokes equations in half space. When the boundary value is located at the subsonic regime, all the possible thirteen asymptotic patterns are classified in \cite{M01}. We consider the most complicated pattern, the superposition of the boundary layer solution, the 1-rarefaction wave, and the viscous 2-shock waves. In this superposition, the boundary layer is degenerate and large. We prove that, if the strengths of the rarefaction wave and shock wave are small, and if the initial data is a small perturbation of the superposition, then the solution asymptotically converges to the superposition up to a dynamical shift for the shock. As a corollary, our result implies the asymptotic stability for the simpler case where the superposition consists of the degenerate boundary layer solution and the viscous 2-shock. Therefore, we complete the study of the asymptotic stability of the inflow problem for the 1D barotropic Navier-Stokes equations for subsonic boundary values.

Paper Structure

This paper contains 29 sections, 21 theorems, 353 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Related results
  4. Ideas of proof
  5. Preliminaries
  6. Boundary layer solution
  7. Rarefaction wave
  8. Viscous shock wave
  9. Estimates on the relative quantities
  10. Poincaré type inequality
  11. A priori estimate and proof of Theorem \ref{['thm:main']}
  12. Local existence
  13. Construction of shift
  14. A priori estimate
  15. Proof of Theorem \ref{['thm:main']}
  16. ...and 14 more sections

Key Result

Theorem 1.1

For any $U_-\in \Omega_{\textup{sub}}$, let $U_*$ be the (unique) constant state which satisfies $U_*\in BL(U_-)\cap \Gamma_{\textup{trans}}$. Then, there exist positive constants $\delta_0$, $\varepsilon_0$ small enough such that for any two constant states $U^*, U_+$ with $U^*\in R_1(U_*)$ and $U_ there exists $\beta>0$ large enough (depending only on the shock strength $|U^*-U_+|$) such that th

Figures (1)

  • Figure 1: (Left) A complete classification of the asymptotic patterns for the inflow problem with $U_-\in \Omega_{\textup{sub}}$. The numbers of classification are the same as those in M01. (Right) The case (13), where the three elementary waves connect $U_-$ and $U_+$. The left-end state $U_-\in \Omega_{\textup{sub}}$ is connected to $U_*\in BL(U_-)\cap\Gamma_{\textup{trans}}$ via the BL solution, $U_*$ is connected to $U^*\in R_1(U_*)$ via the 1-rarefction wave, and $U^*$ is connected to $U_+\in S_2(U^*)$ via the viscous 2-shock wave.

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.1
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 3.1
  • ...and 26 more