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Smooth zero-contact-angle solutions to a thin-film equation around the steady state. (English) Zbl 1159.35039
J. Differ. Equations 245, No. 6, 1454-1506 (2008); corrigendum ibid. 261, No. 2, 1622-1635 (2016).
Summary: 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.

MSC:
35K65 Degenerate parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35K25 Higher-order parabolic equations
35Q35 PDEs in connection with fluid mechanics
35R35 Free boundary problems for PDEs
76A20 Thin fluid films
76D08 Lubrication theory
76D27 Other free boundary flows; Hele-Shaw flows
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