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

Highlights: • There exists a unique steady state. • A high frequency estimate and a numerical Evans function study are performed. • Under linear stability, a steady solution is nonlinearly exponentially stable. -- Abstract: We study existence and stability of steady solutions of the isentropic compressible Navier–Stokes equations on a finite interval with noncharacteristic 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 $\lambda$ (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 $H^2\times H^3$. 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.