Archive for Rational Mechanics and Analysis

, Volume 211, Issue 3, pp 771–818 | Cite as

Upper Bounds on Waiting Times for the Thin-Film Equation: The Case of Weak Slippage

Article

Abstract

We derive upper bounds on the waiting time of solutions to the thin-film equation in the regime of weak slippage \({n\in [2,\frac{32}{11})}\) . In particular, we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary. For \({n\in (2,\frac{32}{11})}\) , our estimates are sharp, for n = 2, they are sharp up to a logarithmic correction term. Note that the case n = 2 corresponds—with a grain of salt—to the assumption of the Navier slip condition at the fluid-solid interface. We also obtain results in the regime of strong slippage \({n \in (1,2)}\) ; however, in this regime we expect them not to be optimal. Our method is based on weighted backward entropy estimates, Hardy’s inequality and singular weight functions; we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long.

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Erlangen-NürnbergErlangenGermany

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