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

• Well-posedness for the thin-film equation with quadratic mobility, zero contact angle.
• Control of the solution's leading-order singular expansion at the free boundary.
• Maximal regularity estimates in weighted Sobolev spaces.

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)(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 L2L2-spaces for the linearized evolution, after suitable subtraction of a(t)+b(t)xβa(t)+b(t)xβ-terms.