Abstract
We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. We obtain sufficient conditions for finite speed of support propagation and for waiting time phenomena by application of a new extension of Stampacchia's lemma for a system of functional equations.
Highlights
The time evolution of thickness of a viscous liquid film spreading over a solid surface under the action of the surface tension and gravity can be described by lubrication models 1–5
The sufficient conditions: h0 x ≤ A|x|4/n for 0 < n < 2, |h0x x | ≤ B|x|4/n−1 for 2 ≤ n < 3, where A and B are some positive constants on nonnegative initial data, h0 for the occurrence of waiting time phenomena were derived by Dal Passo et al 26 for the classic thin film equation: ht |h|nhxxx x 0
The goal is to take δ → 0, ε → 0 in such a way that 1 Tloc,δε → Tloc > 0, 2 the solutions hδε converge to a nonnegative limit, h, which is a generalized weak solution, and 3 h inherits certain a priori bounds. This is done by proving various a priori estimates for hδε that are uniform in δ and ε and hold on a time interval 0, Tloc that is independent of δ and ε
Summary
The time evolution of thickness of a viscous liquid film spreading over a solid surface under the action of the surface tension and gravity can be described by lubrication models 1–5. The main goal of our paper is to study waiting time phenomenon for the coating flows under an assumption of effective slip conditions, that is, we analyze 2.1 that is a modified version of 1.3. The sufficient conditions: h0 x ≤ A|x|4/n for 0 < n < 2, |h0x x | ≤ B|x|4/n−1 for 2 ≤ n < 3, where A and B are some positive constants on nonnegative initial data, h0 for the occurrence of waiting time phenomena were derived by Dal Passo et al 26 for the classic thin film equation: ht |h|nhxxx x 0 These results were based on an energy method developed in 27 for quasilinear parabolic equations. We leave as an open problem the “fast” convection case 0 < n < 1 : finite speed of support propagation and sufficient conditions for waiting time phenomenon
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