Subsonic Flows for the Full Euler Equations in Half Plane

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of subsonic flows governed by the full Euler equations in a half-plane, using a reduction to an elliptic equation and fixed point methods.

Contribution

It introduces a novel approach to analyze subsonic Euler flows in a half-plane by reducing the problem to an elliptic equation and establishing key properties of solutions.

Findings

Existence of subsonic flows in the half-plane proven.

Uniqueness of solutions established under given conditions.

Asymptotic behavior of flows characterized in the far field.

Abstract

We study the subsonic flows governed by full Euler equations in the half plane bounded below by a piecewise smooth curve asymptotically approaching x1-axis. Nonconstant conditions in the far field are prescribed to ensure the real Euler flows. The Euler system is reduced to a single elliptic equation for the stream function. The existence, uniqueness and asymptotic behaviors of the solutions for the reduced equation are established by Schauder fixed point argument and some delicate estimates. The existence of subsonic flows for the original Euler system is proved based on the results for the reduced equation, and their asymptotic behaviors in the far field are also obtained.