Zero surface tension limit of viscous surface waves

TL;DR

This paper proves the zero surface tension limit for viscous incompressible fluid surface waves, establishing local and global results for solutions with small initial data.

Contribution

It introduces a method to rigorously justify the zero surface tension limit for viscous surface waves, including both local and global-in-time results.

Findings

Local strong solutions converge to zero surface tension solutions

Global-in-time convergence for small initial data

Construction of solutions via parabolic regularization

Abstract

We consider the free boundary problem for a layer of viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom and below the atmosphere. For the "semi-small" initial data, we prove the zero surface tension limit of the problem within a local time interval. The unique local strong solution with surface tension is constructed as the limit of a sequence of approximate solutions to a special parabolic regularization. For the small initial data, we prove the global-in-time zero surface tension limit of the problem.