A paralinearization of the 2d and 3d gravity water wave system in infinite depth

Abstract

We consider the 2d and 3d water waves system with gravity and no surface tension in infinite depth. Loss of derivatives from the Dirichlet-to-Neumann operator make studying solutions for long times difficult and until recently only local results were available. Several authors have since made use of paradifferential calculus to overcome these difficulties and prove global regularity in 2d and 3d with and without surface tension. The purpose of this thesis is to formulate the paralinearization of the system based on the Weyl quantization due to Deng-Ionescu-Pausader-Pusateri but with several key modifications. Namely, we work in lower regularity L\(^{2}\)-based Sobolev spaces and do not include surface tension. This makes the problem more difficult by reducing the regularity on the surface elevation. We flatten the interface to arrive at a paralinearization of the Dirichlet-to-Neumann operator. As a result we are able to paralinearize and symmetrize the entire system and derive a single equation for a single complex unknown. The result is suited for obtaining energy estimates that would be useful, for example, when proving rigorous modulation approximations to the water waves in various regimes.