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 $O(\epsilon^{-2})$.

中文翻译：

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具有点涡的二维水波的长时间行为

在本文中，我们研究了流体中具有点涡的二维无粘性不可压缩无限深水波的运动。我们表明，如果点涡足够接近自由边界，泰勒符号条件 $-\frac{\partial P}{\partial \boldmath{n}}\geq 0$ 可能会失败，因此水波可能会受到泰勒不稳定性 假设泰勒符号条件，我们证明了水波系统在 Sobolev 空间中局部适定。此外，我们表明，如果水波是对称的，某个对称涡旋对最初向下行进，那么自由界面将长时间保持光滑，对于大小为 $\epsilon\ll 1$ 的初始数据，寿命为至少 $O(\epsilon^{-2})$。