Two-dimensional gravity waves at low regularity II: Global solutions

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 in Ionescu– Pusateri [Invent. Math. 199 (2015)], Alazard–Delort [Ann. Sci. Éc. Norm. Supér. (4) 48 (2015)], Hunter et al. [Comm. Math. Phys. 346 (2016)], and Ifrim–Tataru [Bull. Soc. Math. France 144 (2016)] 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 Ifrim–Tataru [Ann. Sci. Éc. Norm. Supér. (4) 52 (2019)] in the context of the Benjamin–Ono equations.