複数の国を拠点とする研究協力者たちと連携し、我々は、表面張力が引き起こす薄膜流の研究において生じる方程式の（自己相似）特解の同定に関して、目覚ましい進展を達成した。この研究については、多くの国際的学術誌で論文発表を行った。さらに、近い将来投稿される予定の論文も複数存在する。また、複数の国際会議でも口頭発表を行った

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.

We used a combination of formal asymptotic methods supported by detailed numerical computations in order to investigate solutions to Cauchy-Dirichlet problems for the porous medium equation on a variety of two-dimensional domains.  Such problems can arise physically from consideration of the spreading dynamics of a thin liquid film, where the boundary of the domain acts as an edge over which the film can drain.This research was undertaken as part of an international collaboration with Professor J. R. King (UK) and Professor T. P. Witelski (USA).  This work has been published as:Cauchy-Dirichlet Problems for the Porous Medium Equation, M. Bowen, J. R. King and T. P. Witelski,Disc. Cont. Dyn. Sys., 2023, 43(3&4): 1143-1174. doi: 10.3934/dcds.2022182Special issue celebrating the 75th Birthday of Professor Juan Luis Vazquez.As an outcome of this work, we highlighted a number of open problems that deserve further rigorous analysis.

We have been analytically and computationally investigating the motion of a monolayer thin film (tether) held between two quasi-steady masses.  The monolayer tether is assumed to be stretched by an externally imposed symmetric stagnation point flow and interesting dynamics arise due to the competition between surface tension and convective flow.  In particular, we have been interested in whether the tether breaks in finite time and how such rupture occurs.Mathematical modelling of this phenomena leads to a higher-order non-autonomous nonlinear diffusion-convection equation containing a parameter that takes different values depending on the thickness of the monolayer film.We have analysed the behaviour of the evolution equation for general values of the parameter and have identified that for large enough values of the parameter rupture of the tether does not occur in finite time.We are now writing this research up for publication in the near future.

1) Working with Professor T P Witelski (Duke University, USA), we have been studying out-diffusion solutions of the so-called thin film equation (a fourth order parabolic partial differential equation) on a finite multi-dimensional domain; this extends our recent previous work on the one-dimensional problem.While considering this problem, we decided first to make a preliminary study of the related problem for the lower (second) order porous medium equation.  In this context, we have constructed analytically self-similar solutions that act as large time attractors for solutions defined on sectorial [quarter, half-plane and three-quarter-plane] domains.  We have confirmed these results using numerical simulations.2) While working on this project, I established a new working relationship with Professor L. Smolka (Bucknell University, USA) looking at how thin films evolve in a periodic domain (corresponding physically to the external surface of a cylinder) under the combined effects  of gravity (drainage) and thermal stresses (leading to a non-convex convective flux of fluid).  The interaction of convective effects and surface tension (fourth order parabolic terms) yields solutions containing non-classical shock dynamics, such as undercompressive-compressive shock pairs and undercompressive shocks-rarefaction fans.  We are currently writing up the results of this research for publication in the near future.

Working with Professor T. P. Witekski (Duke University, USA), we mathematically investigated the dynamics of a thin liquid film draining from the edge of a (constrained) domain.  Such problems frequently arise in industrial coating processes where domains are of finite extent.This work has now been published in EJAM (European Journal of Applied Mathematics):Pressure-dipole solutions of the thin-film equationM. BOWEN  and T. P. WITELSKI  European Journal of Applied Mathematicshttps://doi.org/10.1017/S095679251800013XPublished online: 02 April 2018