Degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches

TL;DR

This paper investigates the bifurcation structure of doubly-connected vortex patches in 2D Euler equations, overcoming degeneracy challenges to identify new rotating solutions and addressing an open problem in the field.

Contribution

It introduces a novel bifurcation analysis for degenerate cases of vortex patches, expanding understanding of their rotating solutions beyond previous limitations.

Findings

Families of doubly-connected vortex patches identified

Bifurcation curves constructed via algebraic perturbations

Partially resolves an open problem by Hmidi and Mateu

Abstract

In this paper, we obtain families of two-fold doubly-connected uniformly rotating vortex patches of the 2-D incompressible Euler equations emanating from some specific annuli. The main difficulty comes from strong degeneracy of the problem, neither the kernel of linearization is one-dimensional nor the transeversallity condition holds. To this end, we make a detailed analysis on the nonlinear functional and the bifurcation curves are obtained by perturbing real algebraic varieties defined by truncated polynomials. In addition, our result partially answers an problem proposed by Hmidi and Mateu in \cite{Hmidi2016a} (\emph{Adv.Math.302 (2016), 799-850}).