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 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 < 2d and c₀ < 0, then the solution can be extended after t = T.