Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations

Abstract

This paper is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, we can consider solutions such that the curvature of the initial free surface does not belong to $L^2$. The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and H\older estimates. We first prove tame estimates in Sobolev spaces depending linearly on H\older norms and then we use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these H\older norms.