Time-periodic oscillation of capillary fluid droplet

In this note, we outline the construction of time-periodic oscillating capillary fluid droplet, based on a working paper of the authors. The droplet is modeled by an incompressible, irrotational Eulerian system with free boundary. The key tool developed in the research work is a coordinate-free paradifferential calculus on the compact Lie group SU ( 2 ) , together with its zonal version. The Eulerian free boundary system is paralinearized into a quasilinear dispersive PDE. Despite the presence of small denominators, paradifferential reduction allows one to rewrite the system into a fixed point form balancing all loss of regularity. The construction of time-periodic solutions is then based on bifurcation theory with computer assistance.