Smooth nonradial stationary Euler flows on the plane with compact support

TL;DR

This paper constructs nonradial, compactly supported stationary solutions to the 2D Euler equations, using bifurcation theory, elliptic PDE techniques, and novel regularity estimates, expanding understanding of fluid flows with specific support properties.

Contribution

It introduces a new bifurcation approach for stationary Euler flows with compact support, allowing for nonradial solutions of arbitrary smoothness.

Findings

Existence of nonradial stationary Euler flows with compact support.

Development of sharp regularity estimates for linearized operators.

Overcoming derivative loss via anisotropic weighted functional spaces.

Abstract

We prove the existence of nonradial classical solutions to the 2D incompressible Euler equations with compact support. More precisely, for any positive integer $k$, we construct compactly supported stationary Euler flows of class $C^k(\mathbb{R}^2)$ which are not locally radial. The proof uses a degree-theory-based bifurcation argument which hinges on three key ingredients: a novel approach to stationary Euler flows through elliptic equations with non-autonomous nonlinearities; a set of sharp regularity estimates for the linearized operator, which involves a potential that blows up as the inverse square of the distance to the boundary of the support; and overcoming a serious problem of loss of derivatives by the introduction of anisotropic weighted functional spaces between which the linearized operator is Fredholm.