On the Transition of the Rayleigh-Taylor Instability in 2d Water Waves with Point Vortices

Abstract

In this paper, by considering 2d water waves with a pair of point vortices, we prove the existence of water waves with sign-changing Taylor sign coefficients. That is, the strong Taylor sign condition holds initially, while it breaks down at a later time. Such a phenomenon can be regarded as the transition between the stable and unstable regime in the sense of Rayleigh-Taylor of water waves. As a byproduct, we prove the wellposedness of 2d water waves in Gevrey-2 spaces.