Well-posedness in Sobolev spaces of the full water wave problem in 3-D

Abstract

We consider the motion of the interface of a 3-D inviscid, incompressible, irrotational water wave, with air region above water region and surface tension zero. We prove that the motion of the interface of the water wave is not subject to Taylor instability, as long as the interface separates the whole 3-D space into two simply connected C 2 C^{2} regions. We prove further the existence and uniqueness of solutions of the full 3-D water wave problem, locally in time, for any initial interface that separates the whole 3-D space into two simply connected regions.

Global solutions for the gravity water waves equation in dimension 3

“…The first breakthrough in solving the well posedness for general data came in the work of S. J. Wu [Wu97], [Wu99] who solved the problem in 2 and 3 dimensions. There are many other works on local well posedness.…”

Global Infinite Energy Solutions for the 2D Gravity Water Waves System

“…As proved by Wu [32,33], she showed that, as long as the interface is non-self-intersecting, the Taylor sign condition @p=@n c 0 > 0 always holds for the n-dimensional infinite-depth gravity water waves system, where n 2 (n D 2; 3 for physical relevance). Hence the Taylor instability is not an issue.…”

Seismically generated tsunamis

“…The recent work has depended on establishing the well-posedness of the water wave equations, pioneered by Wu [44,45]. See AlvarezSamaniego & Lannes [46] and their references for more information about the mathematically rigorous validity of these models as approximations of the equations of (inviscid and irrotational) water waves.…”