Quasi-static motion of a capillary drop, II: the three-dimensional case

AbstractA theory is presented to analyze the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia. The theory is developed here on the three-dimensional version of the relevant free-boundary model for Stokes equations. Within this context we show that if the free-boundary initiates close to a spherer=1+ελ0(ω),|ε|small,ω=(θ,ϕ),then there exists a global-in-time solution with free boundaryr=1+λ(ω,t,ε)=1+∑n⩾1λn(ω,t)εn,which approaches a sphere exponentially fast as t→∞. Moreover, we prove that if λ0(ω) is analytic (resp. C∞) in ω, then the velocity u→(x,t,ε), the pressure p(x,t,ε) and the free-boundary λ are all jointly analytic (resp. C∞) in (x,ε). In an earlier paper, we considered the analogous problem for a two-dimensional drop. Although the three-dimensional problem proceeds along similar lines, the analysis is more complicated due to the fact that we work here with spherical harmonics and vector spherical harmonics