An overview on the thin-film equation

A droplet spreading on a glass, in spite of being a very common phenomenon, raises quite a few problems of basic nature whose solution is yet debated: among them, the modeling of the interface which separates "dry" and "wet" regions. The simplest way to explore these issues is to look at the lubrication regime, in which droplets may be modeled by free boundary problems for fourth-order degenerate PDEs, the so-called thin-film equations (TFEs). After briefly reviewing the state of the art and the main open problems for the TFE, I will concentrate on a regularity theory which has been introduced together with Hans Knuepfer and Felix Otto in the case of a linearly degenerate mobility and a zero contact-angle at the free boundary. The TFE is viewed as a classical free boundary problem, and the strategy is based on a-priori energy-type estimates which provide "minimal" conditions on the initial datum under which a unique, global, and smooth solution exists (in case of perturbations of the stationary solution).