Liquid drop with capillarity and rotating traveling 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.
Keywords
Cite
@article{arxiv.2408.02333,
title = {Liquid drop with capillarity and rotating traveling waves},
author = {Pietro Baldi and Vesa Julin and Domenico Angelo La Manna},
journal= {arXiv preprint arXiv:2408.02333},
year = {2026}
}
Comments
Main changes in version 2: Lemma 4.2 has a new, simpler and shorter proof; Section 5 has a simpler, more classical and much shorter proof, and a slightly weaker result; former Section 7 (Appendix) has been removed. Former Sections 5 and 7 of arXiv:2408.02333v1 have been moved, and improved, into a separate article about the Dirichlet-Neumann operator