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.
Copyright. All Rights Reserved.