Centre manifold reduction approach for the lubrication equation

• Published in: Nonlinearity Vol. 24; no. 8; pp. 2347 - 2369
• Main Authors: KITAVTSEV, G, RECKE, L, WAGNER, B
• Format: Journal Article
• Language: English
• Published: Bristol Institute of Physics 01.08.2011
• Subjects: Asymptotic properties; Droplets; Dynamical systems; Dynamics; Exact sciences and technology; Global analysis, analysis on manifolds; Lubrication; Manifolds; Mathematical analysis; Mathematical methods in physics; Mathematics; Ordinary differential equations; Other topics in mathematical methods in physics; Partial differential equations; Physics; Reduction; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Liquid film; Differential equation; Position; Nonlinear analysis; Droplet; Partial differential equation; Thin film; Thickness; Array; Parabolic equation; Reduction method; Manifold; Invariant manifold; Asymptotic approximation; Lubrication; Mathematical physics
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/24/8/010

The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold. The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness epsilon undergo a slow-time coarsening dynamics. With this situation in mind, we construct an asymptotic approximation of the corresponding invariant manifold, that is parametrized by a family of droplet pressures and positions, in the limit when epsilon arrow right 0. The approach is inspired by the paper by Mielke and Zelik (2009 Mem. Am. Math. Soc. 198 1-97), where the centre manifold reduction was carried out for a class of semilinear systems. In this study this approach is considered for quasilinear degenerate parabolic PDEs such as lubrication equations. While it has previously been shown by Glasner and Witelski (2003 Phys. Rev. E 67 016302), that the system of ODEs governing the coarsening dynamics can be obtained via formal asymptotic methods, the centre manifold reduction approach presented here pursues the rigorous justification of this asymptotic limit.