Analysis and PDE Seminar——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 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 $O(\epsilon^{-2})$.