Large time wellposdness to the 3-D Capillary-Gravity Waves in the long wave regime

TL;DR

This paper proves the long-time well-posedness of the 3-D capillary-gravity water wave system in the long wave regime, establishing existence and uniqueness of solutions over extended periods under weakly transverse conditions.

Contribution

It demonstrates the existence and uniqueness of solutions for the 3-D water wave system over long times in the long wave regime, a significant extension of previous short-time results.

Findings

Solutions exist and are unique on $[0, T/ ext{eps}]$ for the water wave system.

The analysis applies to weakly transverse long waves with initial data.

Subsequent work will approximate solutions using decoupled KP equations.

Abstract

In the regime of weakly transverse long waves, given long-wave initial data, we prove that the nondimensionalized water wave system in an infinite strip under influence of gravity and surface tension on the upper free interface has a unique solution on $[0,{T}/\eps]$ for some $\eps$ independent of constant $T.$ We shall prove in the subsequent paper \cite{MZZ2} that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev-Petviashvili (KP) equations.