Long Time Behavior of the 2D Water Waves with Point Vortices
Student No.:100
Time:Tue 15:30-16:30, Dec.4
Instructor:苏庆堂 Qingtang Su (University of Michigan)  
Place:Lecture Hall, Jin Chun Yuan West Bldg.
Starting Date:2018-12-4
Ending Date:2018-12-4

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 point vortices are sufficient close to the free boundary, so the water waves could be subject to Taylor instability. Assuming 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 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 $O(\epsilon^{-2})$.