This memoir 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, the authors 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 Holder estimates. The authors first prove tame estimates in Sobolev spaces depending linearly on Holder norms and then use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these Holder norms.
Introduction
Strichartz estimates
Cauchy problem
Appendix A. Paradifferential calculus
Appendix B. Tame estimates for the Dirichlet-Neumann operator
Appendix C. Estimates for the Taylor coefficient
Appendix D. Sobolev estimates
Appendix E. Proof of a technical result
Bibliography.
T. Alazard, Ecole Normale Superieure, Paris, France.
N. Burq, Universite Paris-Sud, Orsay, France.
C. Zuily, Universite Paris-Sud, Orsay, France.
