Well-posedness for the Navier-slip thin-film equation in the case of complete wetting

Journal article (2014)

Authors

Lorenzo Giacomelli Sapienza University of Rome
M.V. Gnann Max Planck Institute for Mathematics in the Sciences
H. Knüpfer University of Heidelberg
Felix Otto Max Planck Institute for Mathematics in the Sciences
Affiliation
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Lubrication theory Free boundary problems Degenerate-parabolic fourth-order equations Nonlinear parabolic equations Parabolic maximal regularity Thin-film equations
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Published Date

01-07-2014

Language

English

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Abstract

We are interested in the thin-film equation with zero-contact angle and quadratic mobility, modeling the spreading of a thin liquid film, driven by capillarity and limited by viscosity in conjunction with a Navier-slip condition at the substrate. This degenerate fourth-order parabolic equation has the contact line as a free boundary. From the analysis of the self-similar source-type solution, one expects that the solution is smooth only as a function of two variables (x, xβ) (where x denotes the distance from the contact line) with β=13-14≈0.6514 irrational. Therefore, the well-posedness theory is more subtle than in case of linear mobility (coming from Darcy dynamics) or in case of the second-order counterpart (the porous medium equation).Here, we prove global existence and uniqueness for one-dimensional initial data that are close to traveling waves. The main ingredients are maximal regularity estimates in weighted L2-spaces for the linearized evolution, after suitable subtraction of a(t)+b(t)xβ-terms.