Classical solvability to the free boundary problem for a foam drainage equation. I. From the top to the interface

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. [Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza 2, 103–108 (1988)] in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region from the top to the interface in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and the comparison theorem. Moreover, the existence of its steady solution and its stability are discussed.