Uniformly rotating Euler flows with compactly supported velocity

TL;DR

This paper constructs smooth, compactly supported, uniformly rotating solutions to 2D Euler equations that are non-radial, expanding understanding of possible flow structures and their geometric constraints.

Contribution

It introduces the first known smooth, finite-energy, non-radial rotating solutions with compact support for 2D Euler equations.

Findings

Existence of non-radial, compactly supported rotating solutions

Solutions can be small perturbations of radial flows

Rigidity results severely constrain flow geometry

Abstract

For any positive integer $k$, we prove the existence of nontrivial $C^k$-smooth uniformly rotating solutions to the 2D incompressible Euler equations with compact spatial support. These solutions, which can be chosen to be small perturbations of radial flows, are the first example of smooth rotating flows with finite energy which are not locally radial. We also prove new rigidity results for rotating solutions which show that the geometric structure of these flows is severely constrained.