The thin-film equation is a fourth-order degenerate PDE related to the evolution of liquid films or droplets over a solid substrate. In the special case of a linearly degenerate mobility, it may be interpreted as a gradient flow with respect to the Wasserstein metric. In this talk I will concentrate on the regularity of its solutions and of its free boundary (where liquid, solid and vapor phases meet). In a joint work with Hans Knuepfer and Felix Otto, the equation is viewed as a classical free boundary problem. We zoom into the free boundary -where zero slope is prescribed- by looking at perturbations of the stationary solution. Our strategy is based on a-priori energy-type estimates which provide "minimal" conditions on the initial datum under which a unique global solution exists. In addition, we obtain smoothness of both the solution and the free boundary for positive times, and convergenge to the stationary solution for large times.
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