Long time behavior of 2d water waves with point vortices

Abstract

In this paper, we study the motion of the two dimensional inviscid incompressible, infinite depth water waves with point vortices in the fluid. We show that Taylor sign condition $-\frac{\partial P}{\partial \boldmath{n}}\geq 0$ can fail if the point vortices are sufficient close to the free boundary, so the water waves could be subject to the Taylor instability. Assuming the Taylor sign condition, we prove that the water wave system is locally wellposed in Sobolev spaces. Moreover, we show that if the water waves is symmetric with a certain symmetric vortex pair traveling downward initially, then the free interface remains smooth for a long time, and for initial data of size $\epsilon \ll 1$, the lifespan is at least $O({\epsilon }^{-2})$.

arXiv comment76 pages, comments are welcome

References in corpus (3)

• A blow-up criteria and the existence of 2d gravity water waves with angled crests
• On a class of self-similar 2D surface water waves
• Wellposedness and singularities of the water wave equations

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• The quartic integrability and long time existence of steep water waves in 2D
• On the transition of the Rayleigh-Taylor instability in 2d water waves
• Justification of Peregrine soliton from full water waves
• On the existence and instability of solitary water waves with a finite dipole