Uniformly rotating analytic global patch solutions for active scalars

TL;DR

This paper constructs a family of smooth, convex, rotating vortex patch solutions bifurcating from ellipses and disks, extending understanding of rotating solutions in active scalar equations.

Contribution

It introduces new analytic convex rotating solutions bifurcating from ellipses and disks for vortex patch equations, with an adaptable proof of analyticity.

Findings

Existence of analytic convex rotating solutions bifurcating from ellipses.

Analyticity proof applicable to solutions bifurcating from disks for Euler and SQG equations.

Extension of rotating patch solutions understanding in active scalar dynamics.

Abstract

We show that there exists a family of analytic convex global rotating solutions for the vortex patch equations, bifurcating from ellipses. As a byproduct, the analyticity proof can also be adapted to the rotating patch solutions bifurcating from disks (also known as V-states) for both the Euler and the generalized surface quasi-geostrophic equation.