Global well-posedness of the 3-D full water wave problem

TL;DR

This paper proves that small initial disturbances in the 3-D full water wave problem lead to unique, globally existing solutions that decay over time, highlighting the nonlinear nature of the equations.

Contribution

It establishes global well-posedness for the 3-D water wave problem with small initial data, emphasizing the cubic and higher-order nonlinearities involved.

Findings

Existence and uniqueness of solutions for all time

Solutions decay at a rate of 1/t

Nonlinearity is predominantly cubic and higher order

Abstract

We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate $1/t$.