Well-posedness for the Navier-Slip Thin-Film Equation in the Case of Complete Wetting
Lorenzo Giacomelli, Manuel Gnann, Hans Knüpfer, and Felix Otto
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Submission date: 02. Jan. 2014 (revised version: January 2014)
published in: Journal of differential equations, 257 (2014) 1, p. 15-81
DOI number (of the published article): 10.1016/j.jde.2014.03.010
MSC-Numbers: 35K65, 35K35, 35K55, 35Q35, 35R35, 76A20, 76D08
Keywords and phrases: degenerate parabolic equations, higher-order parabolic equations, Nonlinear parabolic equations, thin-film equations, free boundary problems, thin fluid films, lubrication theory
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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 = ≈ 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.