A semi-group approach to the thin-film equation with general mobility

Abstract

This thesis treats the thin-film equation which models the film height $h$ for a viscous film in the complete wetting regime. We show existence and uniqueness to the thin-film equation with mobility m(h) = hⁿ and mobility exponent n∈ (1,3/2)∪ (3/2,3). The thin-film equation is rewritten as an abstract Cauchy problem and usage of semi-group theory yields maximal L^p-regularity for the linearized problem. With a fixed point argument, analogous to the one used by Giacomelli, Gnann, Knüpfer and Otto in their 2014 paper, the nonlinear problem is treated. Under a smallness condition on the initial value to a suitably transformed version of the thin-film equation, we obtain a solution in L^p(0,∞;H_{k-2,α-1/2})∩ \dot{W}^1,p(0,∞;H_{k+2,α+1\2}), where the H-spaces denote weighted Sobolev spaces. The novelty of this work lies in the usage of L^p-spaces in time, where the existing literature only deals with L²-spaces. It is found that the L^p setting allows for treatment of all n∈ (1,3/2)∪ (3/2,3).