On the Cauchy problem for gravity water waves

Thomas Alazard (1) , Nicolas Burq (2) , Claude Zuily (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of~$C^{3/2+\epsilon}$-class for some $\epsilon>0$ and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.

Domaines: Equations aux dérivées partielles [math.AP]

Thomas Alazard, Nicolas Burq, Claude Zuily. On the Cauchy problem for gravity water waves. 2012. ⟨hal-00760452v2⟩