Existence results for the quasistationary motion of a free capillary liquid drop
The authors analyse the multidimensional (space dimension \(N \geq 3\)) Stokes flow driven by surface tension, which is a mathematical model for the quasistationary motion of a viscous liquid drop under the influence of capillary forces. \noindent The problem is given in its dimensionless form by \[ - \triangle u + \nabla p = 0, \quad \text{div} \;u =0 \quad \text{in} \;\Omega(t); \quad {\mathcal T}(u,p) \;n = \kappa \;n \quad \text{on} \;\Gamma(t), \] where \(\Omega(t) \subset \mathbb{R}^N\) is the bounded domain occupied by the drop at the time \(t \geq 0\). Furthermore, \(u\) and \(p\) are the velocity and pressure, \({\mathcal T} = - p I + (\nabla u) + (\nabla u)^T\) is the stress tensor, \(n\) and \(\kappa\) are the outward normal vector and the twice mean curvature of the free surface \(\Gamma(t)\). The problem is completed by a kinematic boundary condition describing the motion of the interface \(\Gamma(t)\). \smallskip\noindent After studying the problem on a fixed domain, the surface motion law is locally reformulated as a fully nonlinear parabolic equation on a smooth manifold. As main statements the authors prove a short-time existence and uniqueness result in Sobolev spaces for general initial domains as well as global existence and exponential decay of the solution near the stable equilibrium solutions given by balls. Main tools for the proofs are, among others, analytic expansions, invariance properties and a priori estimates.
scientific article; zbMATH DE number 1045992