Existence of analytic non-convex V-states

TL;DR

This paper proves the existence of new non-convex V-states with analytic boundaries for the 2D Euler equation, expanding beyond the known circular and elliptical solutions using advanced analysis and computer-assisted methods.

Contribution

It introduces the first known non-convex analytic V-states, demonstrating their existence through a novel combination of linearized operator analysis and computational proofs.

Findings

Existence of non-convex analytic V-states established.

New classes of vortex patches with complex shapes identified.

Methodology combines analytical and computer-assisted techniques.

Abstract

V-states are uniformly rotating vortex patches of the incompressible 2D Euler equation and the only known explicit examples are circles and ellipses. In this paper, we prove the existence of non-convex V-states with analytic boundary which are far from the known examples. To prove it, we use a combination of analysis of the linearized operator at an approximate solution and computer-assisted proof techniques.