Two dimensional gravity waves at low regularity II: Global solutions

TL;DR

This paper advances the understanding of two-dimensional water wave equations by establishing global solutions at minimal regularity and decay assumptions, building on previous high-regularity results.

Contribution

It improves existing global well-posedness results for low regularity water wave solutions in two dimensions, utilizing novel balanced cubic estimates and nonlinear vector field Sobolev inequalities.

Findings

Proved global solutions for small, localized data at minimal regularity.

Extended previous high-regularity results to lower regularity settings.

Introduced nonlinear vector field Sobolev inequalities in this context.

Abstract

This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such solutions have been proved to exist earlier in [15, 7, 10, 12] in much higher regularity. Our goal in this paper is to improve these results and prove global well-posedness under minimal regularity and decay assumptions for the initial data. One key ingredient here is represented by the balanced cubic estimates in our first paper. Another is the nonlinear vector field Sobolev inequalities, an idea first introduced by the last two authors in the context of the Benjamin-Ono equations [14].