Degenerate bifurcations of two-fold doubly-connected vortex patches

Abstract

In this paper we consider the existence of uniformly rotating doubly-connected vortex patch near some annuli. Such annuli correspond to a very degenerate case in the scenario of linearization. Upon a careful analysis on the nonlinear structure of the functional, we prove that there exists a family of two-fold uniformly rotating doubly-connected vortex patch emanating from these annuli. This result partially answers an open problem proposed by Hmidi and Mateu in \cite{Hmidi2016a}