Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions

TL;DR

This paper investigates the existence and stability of steady solutions to the compressible Navier-Stokes equations on a finite interval, demonstrating conditions for spectral and nonlinear exponential stability for various data amplitudes.

Contribution

It establishes the existence, uniqueness, and stability conditions of steady solutions with noncharacteristic boundary conditions for general data amplitudes.

Findings

Existence and uniqueness of steady solutions.

Spectral gap and stability properties identified.

Numerical Evans function analysis supports universal stability.

Abstract

We study existence and stability of steady solutions of the isentropic compressible Navier-Stokes equations on a finite interval with non characteristic boundary conditions, for general not necessarily small-amplitude data. We show that there exists a unique solution, about which the linearized spatial operator possesses (i) a spectral gap between neutral and growing/decaying modes, and (ii) an even number of nonstable eigenvalues ? (with a nonnegative real part). In the case that there are no nonstable eigenvalues, i.e., of spectral stability, we show this solution to be nonlinearly exponentially stable in H2 X H3. Using "Goodman-type" weighted energy estimates, we establish spectral stability for small-amplitude data. For large amplitude data, we obtain high-frequency stability, reducing stability investigations to a bounded frequency regime. On this remaining, bounded-frequency regime, we carry out a numerical Evans function study, with results again indicating universal stability of solutions.