Liquid drop with capillarity and rotating traveling waves

TL;DR

This paper analyzes a 3D liquid drop with capillarity, extending classical flat water wave results to spherical geometry, and proves the bifurcation of rotating traveling wave solutions.

Contribution

It extends classical capillary water wave theory to spherical drops and proves the existence of nontrivial rotating traveling wave solutions.

Findings

Reduction to boundary problem with Hamiltonian structure

Analyticity and tame estimates for Dirichlet-Neumann operator

Existence of bifurcating non-spherical rotating waves

Abstract

We consider the free boundary problem for a 3-dimensional, incompressible, irrotational liquid drop of nearly spherical shape with capillarity. We study the problem from the beginning, extending some classical results from the flat case (capillary water waves) to the spherical geometry: the reduction to a problem on the boundary, its Hamiltonian structure, the analyticity and tame estimates for the Dirichlet-Neumann operator in Sobolev class, and a linearization formula for it, both with the method of the good unknown of Alinhac and by a differential geometry approach. Then we prove the bifurcation of traveling waves, which are nontrivial (i.e., nonspherical) fixed profiles rotating with constant angular velocity.