Liquid drop with capillarity and rotating traveling waves

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.

💡 Research Summary

The paper addresses the free‑boundary problem for a three‑dimensional incompressible, irrotational liquid drop whose dynamics are governed solely by surface tension. Assuming the drop is star‑shaped with respect to the origin, the authors parametrize the moving boundary by a height function h(t, x) on the unit sphere S², i.e. ∂Ωₜ = {(1 + h(t,x)) x : x ∈ S²}. Under the irrotationality condition the velocity field admits a potential Φ, which satisfies Laplace’s equation inside the drop and a Bernoulli‑type equation on the interior. The kinematic and dynamic boundary conditions (normal velocity equals the fluid normal component and pressure equals σ₀ times mean curvature) lead to the definition of a Dirichlet‑Neumann (DN) operator G(h)ψ, where ψ is the trace of Φ on the boundary.

By pulling back the interior equations to the sphere, the authors obtain a coupled system for (h, ψ) (equations (1.16) and (1.18)). This system is shown to be Hamiltonian: the Hamiltonian functional is the sum of kinetic energy and surface‑tension energy, and the symplectic structure coincides with the canonical one on the pair (h, ψ). The reduction to the boundary and the Hamiltonian formulation extend classical water‑wave results from the flat case to the spherical geometry.

A central technical contribution is the derivation of the shape derivative of the DN operator, i.e. an explicit formula for G′(h)