二维点涡水波的长时间行为
Long time behavior of 2d water waves with point vortices
摘要：
本文研究了二维无粘不可压无限深水波在流体中的运动。我们证明了当点涡足够接近自由边界时，Taylor符号条件$-\frac{\partial P}{\partial \boldmath{n}}\geq 0$可能失效，从而水波可能受到Taylor不稳定性的影响。在Taylor符号条件下，证明了水波系统在索伯列夫空间中的局部适定性.此外，我们表明，如果水波是对称的某一对称涡对最初向下传播，那么自由界面保持光滑的很长一段时间，并为初始数据的大小$ε\ll 1$，寿命至少是$O(ε^{-2})$。
英文摘要：
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 $ε\ll 1$, the lifespan is at least $O(ε^{-2})$.
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