Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 265, Issue 11, Pages 5832-5958Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2018.07.015
Keywords
Free boundary; Classical solutions; Moving contact line; Fourth-order degenerate-parabolic equations; Lubrication approximation; Navier slip
Categories
Funding
- Fields Institute for Research in Mathematical Sciences in Toronto
- University of Michigan at Ann Arbor
- Max Planck Institute for Mathematics in the Sciences in Leipzig
- US National Science Foundation [NSF DMS-1054115]
- Deutsche Forschungsgemeinschaft [334362478]
- FSMP fellowship
- EPDI fellowship
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We consider the thin-film equation delta(t)h + del . (h(2) del Delta h) = 0 in physical space dimensions (i.e., one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension), which corresponds to the lubrication approximation of the Navier-Stokes equations of a three-dimensional viscous thin fluid film with Navier-slip at the substrate. This equation can have a free boundary (the contact line), moving with finite speed, at which we assume a zero contact angle condition (complete-wetting regime). Previous results have focused on the 1 + 1-dimensional version, where it has been found that solutions are not smooth as a function of the distance to the free boundary. In particular, a well-posedness and regularity theory is more intricate than for the second-order counterpart, the porousmedium equation, or the thin-film equation with linear mobility (corresponding to Darcy dynamics in the Hele-Shaw cell). Here, we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling-wave profile. This leads to control on the free boundary and in particular its velocity. (C) 2018 Published by Elsevier Inc.
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