Long time behavior of 2d water waves with point vortices

This talk is regarding the mathematical analysis of  long time behavior of two dimensional incompressible inviscid fluids with infinite depth, without surface tension. We refer such fluids as water waves. 

For irrotational case, it is well-known that the water waves remains smooth and small globally in time, under some reasonable smallness and smoothness assumptions on the initial slope and velocity of the fluid-air interface,  see the work by Sijue Wu,  Inonescu-Pusateri,  Alazard-Delort, Ifrim-Hunter-Tataru. Regarding the long time behavior of the rotational water waves, Ifrim and Tataru studied the constant vorticity case and proved cubic lifespan. For water waves with general nonconstant vorticity profiles, it's difficult to keep track of the impact of the vorticity on the free surface, and therefore the study of long time behavior of rotational water waves is largely open. 

Our goal is to study the long time behavior of rotational water waves. In this talk, we'll 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 $O(\epsilon^{-2})$. We'll also discuss how extend to a global result for sufficiently smooth and small localized initial data.