The lifespan of small data solutions in two dimensional capillary water waves

TL;DR

This paper studies the lifespan of small data solutions for two-dimensional capillary water waves without gravity, proving cubic lifespan for small data and global solutions for localized data.

Contribution

It establishes the lifespan estimates for small data solutions and demonstrates global existence for localized initial data in the capillary water wave problem.

Findings

Small data solutions have at least cubic lifespan.

Localized small data solutions lead to global existence.

The analysis is conducted in holomorphic coordinates for the water wave equation.

Abstract

This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.