Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

TL;DR

This paper proves the stability and uniqueness of supersonic contact discontinuities in steady compressible Euler flows within a finite nozzle, using Lagrangian transformations and iterative estimates.

Contribution

It introduces a novel approach to handle free boundary problems in supersonic flows by transforming and estimating solutions in cornered domains.

Findings

Existence of a unique weak solution close to the background state.

The solution features two smooth supersonic flows separated by a smooth contact discontinuity.

The method applies to flows with small perturbations of the incoming flow and nozzle walls.

Abstract

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial-boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.