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
κ of the free surface Σ
t , the trace (
V,
B) of the velocity at the free surface, and the outer normal derivative
\(\frac{{\partial P}}{{\partial n}}\) of the pressure
P satisfy
$$\begin{array}{*{20}c} {\mathop {\sup }\limits_{t \in 0,T]} \left\| {\kappa (t)} \right\|_{L^p \cap L^2 } + \int_0^T {\left\| {(\nabla V,\nabla B)(t)} \right\|_{L^\infty }^6 dt < + \infty ,} } \\ {\mathop {\inf }\limits_{(t,x,y) \in 0,T] \times \sum _t } - \frac{{\partial P}}{{\partial n}}(t,x,y) \geqslant c_0 ,} \\ \end{array} $$
, for some
p < 2
d and c
0 < 0, then the solution can be extended after
t =
T.