Stability of Wave Patterns to the Inflow Problem of Full Compressible Navier-Stokes Equations

TL;DR

This paper investigates the existence and stability of wave patterns, including boundary layers and various waves, in the inflow problem of full compressible Navier-Stokes equations on a half line, using energy methods.

Contribution

It provides new results on the existence and asymptotic stability of boundary layer solutions and wave superpositions in the inflow problem for compressible Navier-Stokes equations.

Findings

Existence and non-existence of boundary layer solutions in different flow regimes.

Asymptotic stability of single contact wave and wave superpositions.

Stability results hold even with large amplitude rarefaction waves.

Abstract

The inflow problem of full compressible Navier-Stokes equations is considered on the half line $(0,+\infty)$. Firstly, we give the existence (or non-existence) of the boundary layer solution to the inflow problem when the right end state $(\rho_+,u_+,\theta_+)$ belongs to the subsonic, transonic and supersonic regions respectively. Then the asymptotic stability of not only the single contact wave but also the superposition of the boundary layer solution, the contact wave and the rarefaction wave to the inflow problem are investigated under some smallness conditions. Note that the amplitude of the rarefaction wave can be arbitrarily large. The proofs are given by the elementary energy method.