Asymptotics for the spectrum of a thin film equation in a singular limit
- Kitavtsev, Georgy
- Recke, Lutz
- Wagner, Barbara
2010 Mathematics Subject Classification
- 76D08 34E057 35B35 35P15 35P20
- spectrum analysis, lubrication equation, asymptotic analysis
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 $eps$.
- SIAM J. Appl. Dyn. Syst., 11(4) (2012), pp. 1425--1457.