Thin viscous films: Thinning driven by surface-tension energy dissipation

Abstract

We study the evolution of a thin film of fluid modeled by the lubrication approximation for thin viscous films. We prove existence of (dissipative) strong solutions for the Cauchy problem when the sub-diffusive exponent ranges between 3/8 and 2; then we show that these solutions tend to zero at rates matching the decay of the source-type self-similar solutions with zero contact angle. Finally, we introduce the weaker concept of dissipative mild solutions and we show that in this case the surface-tension energy dissipation is the mechanism responsible for the H 1 -norm decay to zero of the thickness of the film at an explicit rate. Relaxed problems, with second-order nonlinear terms of porous media type are also successfully treated by the same means.

Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films

“…The existence of nonnegative weak solutions, their qualitative behaviour, and regularity for (3) were rigorously studied in [13][14][15]. The asymptotic analysis results for classical and weak solutions of the thin-film equation were obtained in [16,17]. One of the most well known and still open questions is the uniqueness of strong nonnegative solutions for (3).…”