Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 245, Issue 6, Pages 1454-1506Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2008.06.005
Keywords
existence and uniqueness; fourth order degenerate parabolic equations; thin-film equations; free boundary problems; thin fluid films; lubrication theory; Hele-Shaw flow
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Funding
- SFB 611
- RTN-Programme [HPRN-CT-2002-00274]
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In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the complete wetting regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide minimal conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary. (C) 2008 Elsevier Inc. All rights reserved.
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