Global Solutions for the gravity water waves system: Infinite depth setting and flat bottom setting

Abstract

In this thesis, we study the gravity water waves system in two settings: (i) the water region is two dimensional and there is no bottom, (ii) the water region is three dimensional and the bottom of the water region is flat.

For the $2D$ gravity water waves system in the infinite depth setting, we prove the global existence and the modified scattering for a class of initial data, which can have infinite energy. The initial data is only required to be small above the level $\dot{H}^{1/5}\times \dot{H}^{1/5+1/2}$. No assumption is assumed below this level. As a direct consequence, the momentum condition assumed on the physical velocity in all previous small energy results by Ionescu-Pusateri\cite{IP1}, Alazard-Delort\cite{alazard} and Ifrim-Tataru \cite{tataru3} is removed.

For the $3D$ gravity water waves system in the flat bottom setting, we prove global existence for suitably small initial data and non-existence of traveling waves below a certain level of smallness, which strongly contrasts the behavior of the solution of the same system in the $2D$ case.