Stability of Steady Multi-Wave Configurations for the Full Euler Equations of Compressible Fluid Flow

TL;DR

This paper proves the global stability of steady four-wave configurations, including shocks, vortex sheets, and entropy waves, in supersonic Euler flows under BV perturbations, using a modified Glimm scheme and functional.

Contribution

It develops a new analytical framework and a modified Glimm scheme to establish the global stability of complex multi-wave configurations in Euler equations.

Findings

Steady four-wave configurations are globally stable in supersonic flow.

The Glimm functional decreases due to reflection coefficient properties.

Approximate solutions converge to a global entropy solution.

Abstract

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.