Water waves problem with surface tension in a corner domain I: A priori estimates with constrained contact angle

TL;DR

This paper establishes a priori estimates for 2D water waves with surface tension and a non-zero contact angle in corner domains, showing no singularities occur when the contact angle is less than π/6.

Contribution

It provides the first a priori estimates for water waves with surface tension in corner domains considering contact angles, using elliptic and geometric methods.

Findings

No singularity for contact angles less than π/6

Dissipation occurs at the contact point due to surface tension

Elliptic estimates are effective in corner domain analysis

Abstract

We study the two dimensional water waves problem with surface tension in the case when there is a non-zero contact angle between the free surface and the bottom. In the presence of surface tension, dissipations take place at the contact point. Moreover, when the contact angle is less than $\pi/6$, no singularity appears in our settings. Using elliptic estimates in corner domains and a geometric approach, we prove an a priori estimate for the water waves problem.