On the Cauchy problem for gravity water waves
Résumé
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 $C^{3/2}$ class and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We also take benefit from our low regularity result and an elementary observation to solve a question raised by Boussinesq on the water waves problem in a canal. We reduce the system using a paradifferential approach.
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