We find solutions that describe the levelling of a thin fluid film, comprising a non-Newtonian power-law fluid, that coats a substrate and evolves under the influence of surface tension. We consider the evolution from periodic and localized initial conditions as separate cases. The particular (similarity) solutions in each of these two cases exhibit the generic property that the profiles are weakly singular (that is, higher-order derivatives do not exist) at points where the pressure gradient vanishes. Numerical simulations of the thin film equation, with either periodic or localized initial condition, are shown to approach the appropriate particular solution.

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

The main result of this paper is the proof of uniqueness of non-negative entropy solutions of the thin film equation ht + (|h|nhxxx)x = 0 for < n < 4. The uniqueness proved under assumptions that the initial data satisfy a finite β-entropy condition for some negative enough exponent β and that the solution is locally monotone at the touchdown point. The new dissipated functional recently constructed by Laugesen (Commun. Pure Appl. Anal., 4(3):613–634, 2005) is used to prove an auxiliary energy equality, and then Grönwall's lemma leads to uniqueness.