Break-down criterion for the water-wave equation

Abstract

We study the break-down mechanism of smooth solution for the gravity water-wave equation of infinite depth. It is proved that if the mean curvature $\kappa$ of the free surface $\Sigma_t$, the trace $(V,B)$ of the velocity at the free surface, and the outer normal derivative $\frac {\pa P} {\pa \textbf{n}}$ of the pressure $P$ satisfy \beno &&\displaystyle\sup_{t\in [0,T]}\|\kappa(t)\|_{L^p\cap L^2}+\int_0^T\|(\na V, \na B)(t)\|_{L^\infty}^6dt<+\infty, &\displaystyle\inf_{(t,x,y)\in [0,T]\times \Sigma_t}-\frac {\pa P} {\pa \textbf{n}}(t,x,y)\ge c_0, \eeno for some $p>2d$ and $c_0>0$, then the solution can be extended after $t=T$.