Upper bounds on waiting times for the Thin-film equation: The case of weak slippage

J.L. Fischer, Archive for Rational Mechanics and Analysis 211 (2014) 771–818.

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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 ∈ [2, 32\11). In particular, we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary. For n ∈ (2, 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 ∈ (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.
Publishing Year
Date Published
2014-01-01
Journal Title
Archive for Rational Mechanics and Analysis
Volume
211
Issue
3
Page
771 - 818
IST-REx-ID

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Fischer JL. Upper bounds on waiting times for the Thin-film equation: The case of weak slippage. Archive for Rational Mechanics and Analysis. 2014;211(3):771-818. doi:10.1007/s00205-013-0690-0
Fischer, J. L. (2014). Upper bounds on waiting times for the Thin-film equation: The case of weak slippage. Archive for Rational Mechanics and Analysis, 211(3), 771–818. https://doi.org/10.1007/s00205-013-0690-0
Fischer, Julian L. “Upper Bounds on Waiting Times for the Thin-Film Equation: The Case of Weak Slippage.” Archive for Rational Mechanics and Analysis 211, no. 3 (2014): 771–818. https://doi.org/10.1007/s00205-013-0690-0.
J. L. Fischer, “Upper bounds on waiting times for the Thin-film equation: The case of weak slippage,” Archive for Rational Mechanics and Analysis, vol. 211, no. 3, pp. 771–818, 2014.
Fischer JL. 2014. Upper bounds on waiting times for the Thin-film equation: The case of weak slippage. Archive for Rational Mechanics and Analysis. 211(3), 771–818.
Fischer, Julian L. “Upper Bounds on Waiting Times for the Thin-Film Equation: The Case of Weak Slippage.” Archive for Rational Mechanics and Analysis, vol. 211, no. 3, Springer, 2014, pp. 771–818, doi:10.1007/s00205-013-0690-0.

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