Global regularity for 2d water waves with surface tension

TL;DR

This paper proves global regularity and modified scattering for 2D capillary water waves with surface tension, using a robust energy and dispersive analysis that handles singularities and requires only finite energy perturbations.

Contribution

Develops a new method combining energy estimates and dispersive analysis to handle singularities in 2D water waves with surface tension, allowing for broader initial conditions.

Findings

Global regularity for small localized perturbations

Modified scattering behavior established

New treatment of the Dirichlet-Neumann operator in 2D

Abstract

We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of our analysis is to develop a sufficiently robust method, based on energy estimates and dispersive analysis, which allows us to deal with strong singularities arising from time resonances in the applications of the normal form method and nonlinear scattering. As a result, we are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained.