Convergence to Superposition of Boundary Layer, Rarefaction and Shock for the Inflow Problem of the 1D Navier--Stokes Equations

TL;DR

This paper proves the asymptotic stability of complex superpositions of boundary layers, rarefaction, and shock waves in the inflow problem for 1D Navier-Stokes equations, under small perturbations and wave strengths.

Contribution

It establishes the stability of a superposition of boundary layer, rarefaction, and shock waves, including degenerate cases, for the 1D Navier-Stokes inflow problem in subsonic regimes.

Findings

Solution converges to the superposition with a dynamical shift for the shock.

Asymptotic stability is proven for superpositions involving degenerate boundary layers.

Completes the stability analysis for subsonic inflow boundary conditions.

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.