We have undertaken foundational research in order to prepare for a study of the thin film equation on two-dimensional domains.  The focus of this work was to establish the viability of the proposed analytical/numerical methodology and to identify complexities associated with the analysis of higher-order parabolic partial differential equations.

This research consequently followed two separate paths.  

Firstly, developing a research collaboration with Professor T. P. Witelski at Duke University, USA, we have investigated solutions of the lower order porous medium equation on two-dimensional domains.  The knowledge gained from this study will be invaluable in investigating solutions to the higher-order thin-film equation.  In particular we have identified various self-similar solution dynamics (building on previous joint work with Professor J. R. King at the University of Nottingham, UK) and have investigated transitions between the different similarity solution forms.  We are currently preparing this work for publication in the near future.

One of the main differences between the lower order porous medium equation and the higher order thin film equation is the lack of a maximum principle for the latter.  Consequently, solutions of the thin film equation that are initially positive everywhere can become zero at some later time.  Such behaviour has important consequences both for the analysis and computation of solutions, and for physical applications (where the solution becoming zero corresponds to rupture of the thin film).  The conditions under which zeros can form in the solution to the (fourth-order parabolic) thin film equation remains a fundamental open question.

In the second path of research, working with Professor J. R. King, we have completed an investigation into this open question studying (primarily) the symmetric rupture of (one-dimensional) thin films.  This work is to be published shortly in a theme issue of the Philosophical Transactions of the Royal Society A.

We are now in a position to extend the methodology employed in the first path of research to the two-dimensional thin film problems.  As part of the future research, we will require extensions of the results from the second path of research to the two-dimensional case.