Variational structure and two-dimensional subsonic jet flows for compressible Euler system with general incoming flows

TL;DR

This paper establishes the well-posedness of two-dimensional subsonic jet flows in the Euler system with general incoming flows, leveraging a variational structure of the stream function formulation to analyze a Bernoulli free boundary problem.

Contribution

It introduces a variational framework for analyzing subsonic jet flows with vorticity, extending existing methods to flows with nontrivial vorticity and general incoming velocities.

Findings

Proved well-posedness of subsonic jet flows for general incoming flows.

Identified a variational structure in the flow problem.

Demonstrated smallness of inhomogeneous terms near the free boundary.

Abstract

In this paper, we proved the well-posedness theory of compressible subsonic jet flows for two-dimensional steady Euler system with {\it 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 to analyze 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.