On the trivial solutions for the rotating patch model

TL;DR

This paper proves that under certain conditions, the only rotating patch solutions for the Euler equations are Rankine vortices, highlighting their uniqueness among slightly convex domains and at specific angular velocities.

Contribution

It establishes the uniqueness of Rankine vortices as solutions in slightly convex domains and at angular velocity 1/2, using the moving plane method and geometric analysis.

Findings

Rankine vortices are the only solutions in slightly convex domains.

At angular velocity 1/2, the set of V-states is trivial and consists solely of Rankine vortices.

The results hold without geometric restrictions at angular velocity 1/2.

Abstract

In this paper we study the clockwise simply connected rotating patches for Euler equations. By using the moving plane method we prove that Rankine vortices are the only solutions to this problem in the class of {\it slightly} convex domains. We discuss in the second part of the paper the case where the angular velocity $\Omega=\frac12$ and we show without any geometric condition that the set of the V-states is trivial and reduced to the Rankine vortices.