A Classification Theorem for Steady Euler Flows

Abstract

Fix a bounded, analytic, and simply connected domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^2.$ We show that all analytic steady states of the Euler equations with stream function $\ddot{I}ˆ$ are either radial or solve a semi-linear elliptic equation of the form $\hat{I}”\ddot{I}ˆ=F(\ddot{I}ˆ)$ with Dirichlet boundary conditions. In particular, if $\hat{I}\text{c◯}$ is not a ball, then there exists a one to one correspondence between analytic steady states of the Euler equations and analytic solutions of equations of the form $\hat{I}”\ddot{I}ˆ=F(\ddot{I}ˆ)$ with Dirichlet boundary conditions.