# On an Elliptic Free Boundary Problem and Subsonic Jet Flows for a Given Surrounding Pressure

## TL;DR

This paper studies subsonic jet flows in a convergent nozzle governed by a free boundary elliptic PDE, establishing existence, uniqueness, and regularity results depending on nozzle length and flow parameters.

## Contribution

It introduces a well-posedness framework for subsonic jet flows with free boundaries, identifying conditions for existence and uniqueness based on flow speed and nozzle length.

## Key findings

- Existence of a unique subsonic jet flow for certain nozzle lengths
- Identification of a minimal speed and velocity potential difference for flow space
- Optimal regularity and properties of the flows

## Abstract

This paper concerns compressible subsonic jet flows for a given surrounding pressure from a two-dimensional finitely long convergent nozzle with straight solid wall, which are governed by a free boundary problem for a quasilinear elliptic equation. For a given surrounding pressure and a given incoming mass flux, we seek a subsonic jet flow with the given incoming mass flux such that the flow velocity at the inlet is along the normal direction, the flow satisfies the slip condition at the wall, and the pressure of the flow at the free boundary coincides with the given surrounding pressure. In general, the free boundary contains two parts: one is the particle path connected with the wall and the other is a level set of the velocity potential. We identify a suitable space of flows in terms of the minimal speed and the maximal velocity potential difference for the well-posedness of the problem. It is shown that there is an optimal interval such that there exists a unique subsonic jet flow in the space iff the length of the nozzle belongs to this interval. Furthermore, the optimal regularity and other properties of the flows are shown.