Existence Results for the Quasistationary Motion of a Free Capillary Liquid Drop

Abstract

We consider instationary creeping flow of a viscous liquid drop with free boundary driven by surface tension. This yields a nonlocal surface motion law involving the solution of the Stokes equations with Neumann boundary conditions given by the curvature of the boundary. The surface motion law is locally reformulated as a fully nonlinear parabolic (pseudodifferential) equation on a smooth manifold. Using analytic expansions, invariance properties, and a priori estimates we give, under suitable presumptions, a short-time existence and uniqueness proof for the solution of this equation in Sobolev spaces of sufficiently high order. Moreover, it is shown that if the initial shape of the drop is near the ball then the evolution problem has a solution for all positive times which exponentially decays to the ball.

Existence results for Hele–Shaw flow driven by surface tension

“…Finally, global existence of the solution in time and exponential decay of this solution near equilibrium are shown. The problem of Stokes flow driven by surface tension can be treated in completely analogous manner [11].…”

On Stokes flow driven by surface tension in the presence of a surfactant

“…Let us compare the approaches and the resulting evolution equations for the cases γ = const., as discussed in [10], with the case γ = σ(ρ) (and purely convective surfactant transport), as discussed here. (For simplicity, we assume (5.2).)…”

“…The two-dimensional version of this problem has been discussed in e.g. [3,4,12,14], and short-time solvability in the general case was proved in [10,15].…”

Analyticity of Solutions to Nonlinear Parabolic Equations on Manifolds and an Application to Stokes Flow

“…This follows from the special structure of the right hand side and can be proved easily using the weak formulation of (5.2). The details are given in [16], Lemma 1(ii), for (µ, w) = (0, 0), the general case is analogous. It easily follows from Lemma 5.1 that …”

“…Short-time existence and uniqueness results for (5.1)-(5.3) holding in arbitrary spatial dimensions are given in [16] and [22], in [16] it is also shown that the free boundary is C ∞ -smooth for all positive times for which the solution exists, even if Γ(0) has finite smoothness only. For m = 1, see also [5,6,14,20].…”