John, Dominik: Uniqueness and Stability near Stationary Solutions for the Thin-Film Equation in Multiple Space Dimensions with Small Initial Lipschitz Perturbations. - Bonn, 2013. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-33527
@phdthesis{handle:20.500.11811/5764,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-33527,
author = {{Dominik John}},
title = {Uniqueness and Stability near Stationary Solutions for the Thin-Film Equation in Multiple Space Dimensions with Small Initial Lipschitz Perturbations},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2013,
month = dec,

note = {In any number of space variables, we study the Cauchy problem related to the thin-film equation in the simplest case of a linearly degenerate mobility. This equation, derived from a lubrication approximation, also models the surface-tension dominated flow of a thin viscous film in the Hele-Shaw cell.
Our focus is on uniqueness of weak solutions in the complete wetting regime, when a zero contact angle between liquid and solid is imposed. In this case, we transform the problem by zooming into the free boundary and look at small Lipschitz perturbations of a quadratic stationary solution. In the perturbational setting, the main difficulty is to construct scale invariant function spaces based on time-space cylinders. Here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide optimal conditions on the initial data under which a unique solution exists. In fact, this solution can be used to define a class of functions in which the original initial value problem has a unique (weak) solution. Moreover, we show that the (moving) interface between empty and occupied regions is an analytic hypersurface in time and space.},

url = {http://hdl.handle.net/20.500.11811/5764}
}

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