Classical solvability to the two-phase free boundary problem for a foam drainage equation

The paper is devoted to the study of two-phase free boundary problem for nonlinear partial differential equations describing the evolution of a foam drainage in the one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of a liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. In a series of papers, the authors have already solved the same problems without free boundary and with free boundary situated at the lower and the upper parts in the foam column, respectively. In this paper it is shown that the free boundary problem for the foam drainage equations with a sharp interface between dry and wet foams admits a unique global-in-time classical solution; this is done by a standard classical mathematical method, the maximum principle, and the comparison theorem. Moreover, the existence of the steady solution and its stability are shown.