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

  • Julian Fischer


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.


Free Boundary Contact Line Differential Inequality Degenerate Parabolic Equation Porous Medium Equation 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Erlangen-NürnbergErlangenGermany

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