On the transition of the Rayleigh-Taylor instability in 2d water waves

In this paper we prove the existence of water waves with sign-changing Taylor sign coefficients, that is, the strong Taylor sign holds initially, while breaks down at a later time, and vice versa. 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 sharp wellposedness of 2d water waves in Gevrey-2 spaces.

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