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


Author(s): Fischer, Julian
Article Title: Upper bounds on waiting times for the Thin-film equation: The case of weak slippage
Affiliation
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.
Journal Title: Archive for Rational Mechanics and Analysis
Volume: 211
Issue 3
ISSN: 1432-0673
Publisher: Springer  
Date Published: 2014-01-01
Start Page: 771
End Page: 818
DOI: 10.1007/s00205-013-0690-0
Open access: no