Inaugural Conference of the Fibonacci Laboratory

# On the Cauchy Problem for Water Gravity Waves

speaker: Thomas Alazard (CNRS, École Normale Supérieure)

abstract: The main result of this talk clarify the Cauchy theory of the gravity water waves equations (without surface tension) as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). In particular, the initial surfaces we consider turn out to be only of $$C^{3/2}$$ class and consequently have unbounded curvature. We also take benefit from our low regularity result and an elementary observation to solve a question raised by Boussinesq on the water-wave system in a canal.

After suitable para-linearizations, we show that the system can be arranged into an explicit symmetric system of quasilinear waves equation type. As an 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. This is a joint work with Nicolas Burq and Claude Zuily.

timetable:
Wed 7 Mar, 17:25 - 18:25, Sala Azzurra
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