(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Kitavtsev, Georgy; Recke, Lutz; Wagner, Barbara
In this paper the linear stability properties of the steady states of a
no-slip lubrication equation are studied. The steady states are
configurations of droplets and arise during the late-phase dewetting process
under the influence of both destabilizing van der Waals and stabilizing Born
intermolecular forces, which in turn give rise to the minimum thickness eps
of the remaining film connecting the droplets. The goal of this paper is to
give an asymptotic description of the eigenvalues and eigenfunctions of the
problem, linearized about the one-droplet solutions, as epsto 0. For this
purpose, corresponding asymptotic eigenvalue problems with piecewise constant
coefficients are constructed, such that their eigenvalue asymptotics can be
determined analytically. A comparison with numerically computed eigenvalues
and eigenfunctions shows good agreement with the asymptotic results and the
existence of a spectrum gap to a single exponentially small eigenvalue for
sufficiently small eps.
