一个爆破准则与二维斜顶重力水波的存在性
A blow-up criteria and the existence of 2d gravity water waves with angled crests
摘要：
我们考虑二维重力水波方程的制度，包括自由表面与角波峰。我们假设流体是无粘性的，不可压缩的和无旋的，空气密度为零，我们忽略表面张力。在\cite{kw}中，证明了在这个区域中，只有退化的泰勒不等式$-\frac{\partial P}{\partial\bold{n}}\ge 0$成立，在奇点处退化;构造了能量泛函$\frak E$，并证明了先验估计。在本文中，我们表明，一个（广义）的水波动方程的解决方案与光滑的数据将保持光滑，只要$\frak E(t)$保持有限的，并为任何数据满足$\frak E(0)<\infty$，方程是可解的局部时间，一个周期只取决于$\frak E(0)$。
英文摘要：
We consider the two dimensional gravity water wave equation in the regime that includes free surfaces with angled crests. We assume that the fluid is inviscid, incompressible and irrotational, the air density is zero, and we neglect the surface tension. In \cite{kw} it was shown that in this regime, only a degenerate Taylor inequality $-\frac{\partial P}{\partial\bold{n}}\ge 0$ holds, with degeneracy at the singularities; an energy functional $\frak E$ was constructed and an aprori estimate was proved. In this paper we show that a (generalized) solution of the water wave equation with smooth data will remain smooth so long as $\frak E(t)$ remains finite; and for any data satisfying $\frak E(0)<\infty$, the equation is solvable locally in time, for a period depending only on $\frak E(0)$.
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