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 geometric approach. Then, also thanks to the analyticity of the operators involved, we prove the bifurcation of traveling waves, which are nontrivial (i.e., nonspherical) fixed profiles rotating with constant angular velocity. To the best of our knowledge, this is the first example of global-in-time nontrivial solutions of the free boundary problem for the capillary liquid drop.