Surfactants on thin films play a role in biological applications and in paints. Recent work by Grotberg and his collaborators concerns the spreading of surfactants under the influence of the Marangoni force, i.e. the tangential force on the free surface generated by the concentration dependence of the surface tension. This Marangoni force creates a tendency for the surfactant to spread out, leading to a diffusion which is present even if molecular diffusion is neglected.
From a mathematical point of view, the equations lead to a parabolic-hyperbolic system, and the parabolic part degenerates if molecular diffusion is neglected and either the film thickness or the surfactant concentration tends to zero. The hyperbolic part is genuinely nonlinear, allowing for shock waves, only if molecular diffusion is included.
In Renardy's paper, a singularly perturbed problem is considered where, in addition to Marangoni and diffusion effects, small terms resulting from gravity, mean surface tension and van der Waals forces are included in the equations (the first two are formally of higher order as the film thickness tends to zero, and van der Waals forces are small unless the film is extremely thin). If gravity dominates over van der Waals forces, then these higher order terms have a stabilizing effect, regularizing the system to a parabolic problem. The paper considers the continuous dependence of solutions as the regularizing terms tend to zero and the existence of shock structures. In the last paper, the case where the parabolic part of the system degenerates in addressed. While there is an extensive literature on degenerate parabolic equations, the coupling to a hyperbolic equation is a new feature, which had to be addressed by developing a new type of energy estimates. The degeneracy in the parabolic part occurs if either the film height or the surfactant concentration is zero, and solutions with ``fronts" of both kinds are considered. The well-posedness of the system for appropriate classes of initial data is established.