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

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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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  1. Department of Mathematics, University of Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Julian Fischer

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Communicated by F. Otto

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Fischer, J. Upper Bounds on Waiting Times for the Thin-Film Equation: The Case of Weak Slippage. Arch Rational Mech Anal 211, 771–818 (2014). https://doi.org/10.1007/s00205-013-0690-0

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  • DOI: https://doi.org/10.1007/s00205-013-0690-0

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