Well‐posedness for the Navier Slip Thin‐Film equation in the case of partial wetting

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Abstract
Consider a viscous liquid droplet spreading on a surface. The classical slip condition at the liquid‐solid interface is the no‐slip condition. However, this condition yields infinite dissipation rate when the contact line moves (“no‐slip paradox”). For this reason other slip conditions such as the Navier slip condition have been proposed. We prove well‐posedness for a reduced 1‐D fluid model related to Navier slip. It turns out that the profile of the droplet cannot be described by a smooth function (not even for an initially smooth profile). However, existence and uniqueness can be proved in larger classes of spaces that allow for certain classes of singular expansions at the moving contact point.

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