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MiS Preprint
1/2014
Well-posedness for the Navier-Slip Thin-Film Equation in the Case of Complete Wetting
Lorenzo Giacomelli, Manuel Gnann, Hans Knüpfer and Felix Otto
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 $(x,x^\beta)$ (where $x$ denotes the distance from the contact line) with $\beta=\frac{\sqrt{13}-1}{4}\approx 0.6514$ 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 $L^2$-spaces for the linearized evolution, after suitable subtraction of $a(t)+b(t)x^\beta$-terms.
