Variational Structure and Two-Dimensional Subsonic Jet Flows for Compressible Euler System with General Incoming Flows

Abstract

In this paper, we prove the well-posedness theory of compressible subsonic jet flows for a two-dimensional steady Euler system with general incoming horizontal velocity as long as the flux is larger than a critical value. One of the key observations is that the stream function formulation for two-dimensional compressible steady Euler system enjoys a variational structure even when the flows have nontrivial vorticity, so that the jet problem can be reformulated as a domain variation problem. This variational structure helps to adapt the framework developed by Alt, Caffarelli, and Friedman to study the jet problem, which is a Bernoulli-type free boundary problem. A major technical point for analyzing the jet flows is that the inhomogeneous terms in the rescaled equation near the free boundary are always small, even when the vorticity of the flows is big.