Steady self-similar inviscid flow

TL;DR

This paper analyzes steady, self-similar solutions to the 2D compressible Euler equations, characterizing their structure near constant supersonic flows and establishing uniqueness and regularity results for Riemann problem solutions.

Contribution

It provides a detailed characterization of steady self-similar solutions in 2D Euler flows and proves uniqueness and BV regularity for Riemann problem solutions within certain classes.

Findings

Solutions near a constant supersonic background are characterized.

Uniqueness of 1D Riemann problem solutions in small L^ functions.

Backward-in-time Riemann solutions are necessarily BV.

Abstract

We consider solutions of the 2-d compressible Euler equations that are steady and self-similar. They arise naturally at interaction points in genuinely multi-dimensional flow. We characterize the possible solutions in the class of flows L^\infty-close to a constant supersonic background. As a special case we prove that solutions of 1-d Riemann problems are unique in the class of small L^\infty functions. We also show that solutions of the backward-in-time Riemann problem are necessarily BV.