Classical solvability to the initial boundary value problem for a forced foam drainage equation

Abstract

In this paper we study a nonlinear partial differential equation describing the evolution of a forced foam drainage in one dimensional case. A foam drainage equation was proposed by Goldfarb et al. [5] in 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. Mathematical studies on a foam drainage equation so far are restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. This paper is aimed at proving that the initial boundary value problem for the forced foam drainage equation admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and comparison theorem. Moreover, the existence of its steady solution and its stability are shown. This is the first rigorous mathematical result for a series of researches on this equation.