Bifurcation from multiple eigenvalues of rotating traveling waves on a capillary liquid drop
Pietro Baldi, Domenico Angelo La Manna, Giuseppe La Scala
TL;DR
The paper analyzes rotating traveling waves on capillary liquid drops near spherical geometry, with the angular velocity $\omega$ as the bifurcation parameter. It proves that every eigenvalue of the linearized problem, including those of multiplicity, is a bifurcation point, extending known results from simple to multiple eigenvalues. The authors formulate the problem variationally in a Hamiltonian setting, using angular momentum as the action and exploiting a Noether-based transversality to handle a two-dimensional degeneracy from symmetries. A Lyapunov–Schmidt reduction yields a reduced, constrained finite-dimensional bifurcation problem, from which at least $n$ distinct rotating-wave branches emanate for small angular momentum $a$, with intricate symmetry-related multiplicities discussed in the symmetric cases. The work advances the understanding of shape- and velocity-compatible rotating drops and provides a robust framework that combines Hamiltonian structure, variational methods, and group actions to treat multiplicity in bifurcation problems with free boundaries.
Abstract
We consider the free boundary problem for a liquid drop of nearly spherical shape with capillarity, and we study the existence of nontrivial (i.e., non spherical) rotating traveling profiles bifurcating from the spherical shape, where the bifurcation parameter is the angular velocity. We prove that every eigenvalue of the linearized problem is a bifurcation point, extending the known result for simple eigenvalues to the general case of eigenvalues of any multiplicity. We also obtain a lower bound on the number of bifurcating solutions. The proof is based on the Hamiltonian structure of the problem and on the variational argument of constrained critical points for traveling waves of Craig and Nicholls (2000, SIAM J. Math. Anal. 32, 323-359), adapted to the nearly spherical geometry; in particular, the role of the action functional is played here by the angular momentum with respect to the rotation axis. Moreover, the bifurcation equation presents a 2-dimensional degeneration, related to some symmetries of the physical problem. This additional difficulty is overcome thanks to a crucial transversality property, obtained by using the Hamiltonian structure and the prime integrals corresponding to those symmetries by Noether theorem, which are the fluid mass and the component along the rotation axis of the velocity of the fluid barycenter.