Rigidity results for finite energy solutions to the stationary 2D Euler equations

TL;DR

This paper establishes rigidity results for finite energy solutions to the stationary 2D Euler equations, showing under certain conditions that stream functions satisfy specific elliptic equations and streamlines are concentric circles.

Contribution

It proves new rigidity theorems for stationary 2D Euler solutions with finite energy, connecting the structure of solutions to elliptic equations and streamline geometry.

Findings

Stream functions satisfy autonomous semilinear elliptic equations.

Under conditions on vorticity, streamlines are concentric circles.

Energy estimates at infinity are used to derive rigidity results.

Abstract

In this paper we prove rigidity results for classical solutions to the stationary 2D Euler equations in $\mathbb{R}^2$. Assuming that the velocity field has finite energy and that the stagnation set is connected, we prove that the corresponding stream function solves an autonomous semilinear elliptic equation. Under some extra conditions on the vorticity near infinity we can also prove that the streamlines are concentric circles. The proofs include several energy estimates on the behavior of the stream function at infinity, as well as an adaptation of the continuous Steiner symmetrization to our setting.