A remark on the two dimensional water wave problem with surface tension

TL;DR

This paper introduces a simplified reduction of the 2D water wave problem with surface tension into a quasilinear system involving interface tangent angle and velocity difference, and establishes an energy inequality for periodic waves.

Contribution

It proposes a novel, simplified method to reduce the water wave problem to a quasilinear system using only differentiation and complex analysis techniques.

Findings

Derived a new reduction method involving tangent angle and velocity difference.

Established a priori energy inequality for periodic wave solutions.

Simplified the mathematical analysis of the water wave problem with surface tension.

Abstract

We consider the motion of a two-dimensional interface between air (above) and an irrotational, incompressible, inviscid, infinitely deep water (below), with surface tension present. We propose a new way to reduce the original problem into an equivalent quasilinear system which are related to the interface's tangent angle and a quantity related to the difference of tangential velocities of the interface in the Lagrangian and the arc-length coordinates. The new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts. Then if assuming that waves are periodic, we establish a priori energy inequality.