Low regularity Cauchy theory for the water-waves problem: canals and swimming pools

Abstract

The purpose of this talk is to present some recent results about the Cauchy theory of the gravity water waves equations (without surface tension). In particular, we clarify the theory as well in terms of regularity indexes for the initial conditions as fin terms of smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developed in [1, 2], after suitable para-linearizations, the system can be arranged into an explicit symmetric system of quasilinear waves equation type, and consequently can be solved at the usual levels of regularity (initial data in H s ,s>1+d/2). In particular, the system can be solved for initial surfaces having undounded curvature. As another illustration of this reduction, we show that in fact following the analysis by Bahouri-Chemin and Tataru for quasi-linear wave equations, using Strichartz estimates, the regularity threshold can be further lowered, which allows to obtain well posedness for non lipschitz initial velocity fields. We also take benefit from our low regularity result and an elementary (though seemingly yet unknown) observation to solve a question raised by Boussinesq on the water-wave system in a canal.