Recent advances on the global regularity for irrotational water waves

TL;DR

This paper reviews recent mathematical progress on the long-term regularity and global existence of solutions to water wave equations, covering both gravity and capillary effects in two and three dimensions.

Contribution

It provides a unified overview of recent advances in proving global regularity for water waves, including new results on gravity-capillary models and singularity formation.

Findings

Global existence results for 2D and 3D gravity waves

Global regularity for 3D gravity-capillary water waves

Discussion on singularity formation in water wave solutions

Abstract

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach, as in [7, 1, 2]. We then describe in a unified framework, using the Eulerian formulation, global existence results for three dimensional and two dimensional gravity waves, see [70, 146, 145, 87, 5, 6, 79, 80, 136], and our joint result with Deng and Pausader [60] on global regularity for the 3D gravity-capillary model. We conclude this review with a short discussion about the formation of singularities, and give a few additional references to other interesting topics in the theory.