Centre manifold reduction approach for the lubrication equation

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 e 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 -> 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.