Well-posedness of the water-wave problem with surface tension

TL;DR

This paper proves the local well-posedness of the water wave problem with surface tension in finite depth and shows convergence to the no-surface-tension case as surface tension vanishes.

Contribution

It establishes well-posedness in the Eulerian setting for finite depth and analyzes the zero surface tension limit.

Findings

Proves local well-posedness with surface tension

Shows convergence to no-surface-tension solutions as surface tension approaches zero

Works in the Eulerian framework for finite depth

Abstract

In this paper, we prove the local well-posedness of the water wave problem with surface tension in the case of finite depth by working in the Eulerian setting. For the flat bottom, as surface tension tends to zero, the solution of the water wave problem with surface tension converges to the solution of the water wave problem without surface tension.