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International Journal of Differential Equations
Volume 2016 (2016), Article ID 4063740, 11 pages
http://dx.doi.org/10.1155/2016/4063740
Research Article

Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films

1Institute of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USA
2Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, USA

Received 7 March 2016; Accepted 5 April 2016

Academic Editor: Jingxue Yin

Copyright © 2016 Matthew Michal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For the thin-film model of a viscous flow which originates from lubrication approximation and has a full nonlinear curvature term, we prove existence of nonnegative weak solutions. Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipation of the solution arc length that is a well known property for a Hele-Shaw flow. For the classical thin-film model with linearized curvature term, under some restrictions on parameter and gradient values, we also prove analytically the arc length dissipation property for positive solutions. We compare the numerical solutions for both models, with nonlinear and with linearized curvature terms. In regimes when solutions develop finite time singularities, we explain the difference in qualitative behaviour of solutions.