The thin film equation with non zero contact angle: A singular perturbation approach

In this paper we prove the existence of weak solutions for the thin film equation with prescribed non zero contact angle and for a large class of mobility coefficients, in dimension 1. The existence of weak solutions for this degenerate parabolic fourth order free boundary problem was proved by F. Otto in [35] when the mobility coefficient is given by f(u) = u, using a particular gradient flow formulation which does not seem to generalize to other mobility coefficients. Short time existence (and uniqueness) of strong solutions was recently proved by H. Knüpfer and N. Masmoudi in [33, 34] for f(u) = u and f(u) = u and for regular enough initial data (corresponding to a single droplet). In this paper, we use a different approach to prove the global in time existence of weak solutions without condition on the support, by using a diffuse approximation of the free boundary condition. This approach, which can be physically motivated by the introduction of singular disjoining/conjoining pressure forces has been suggested in particular by Bertsch, Giacomelli and Karali in [11]. Our main result is the existence of some weak solutions for the free boundary problem when the mobility coefficient satisfies f(u) ∼ u as u → 0 for some n ∈ [1, 2).