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We consider the spreading of a thin two-dimensional droplet on a solidsubstrate. We use a model for viscous fluids where the evolution is governed byDarcy's Law. At the triple point where air and liquid meet the solid substrate,the liquid assumes a constant, non-zero contact angle ({\it partial wetting}).We show local and global well-posedness of this free boundary problem in thepresence of the moving contact point. Our estimates are uniform in the contactangle assumed by the liquid at the contact point. In the so-called lubricationapproximation (long-wave limit) we show that the solutions converge to thesolution of a one-dimensional degenerate parabolic fourth order equation whichbelongs to a family of thin-film equations. The main technical difficulty is todescribe the evolution of the non-smooth domain and to identify suitable spacesthat capture the transition to the asymptotic model uniformly in the smallparameter $\eps$.

Darcy's flow with prescribed contact angle -- Well-posedness and lubrication approximation