Large-time Behavior of Solutions to the Inflow Problem of Full Compressible Navier-Stokes Equations

TL;DR

This paper analyzes the long-term behavior of solutions to the inflow problem for the full compressible Navier-Stokes equations on a half line, demonstrating the stability of a complex wave structure involving four different wave patterns.

Contribution

It provides a rigorous proof of the asymptotic stability of a superposition of four wave types in the inflow problem, using elementary energy methods.

Findings

Proves stability of the wave superposition under smallness conditions.

Describes the wave structure including boundary layer and rarefaction waves.

Addresses degeneracies and wave interactions in the stability analysis.

Abstract

Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations is investigated on the half line $R^+ =(0,+\infty)$. The wave structure which contains four waves: the transonic(or degenerate) boundary layer solution, 1-rarefaction wave, viscous 2-contact wave and 3-rarefaction wave to the inflow problem is described and the asymptotic stability of the superposition of the above four wave patterns to the inflow problem of full compressible Navier-Stokes equations is proven under some smallness conditions. The proof is given by the elementary energy analysis based on the underlying wave structure. The main points in the proof are the degeneracies of the transonic boundary layer solution and the wave interactions in the superposition wave.