Undercompressive Shocks in Thin Film Flow

Andrea L. Bertozzi, Dept. of Mathematics, Duke University

Nonlinear hyperbolic conservation laws have solutions with propagating `shocks' or discontinuities. Compressive shocks satisfy an `entropy condition' in which characteristics enter the shock on each side. Undercompressive shocks violate this condition. We show that scalar laws with non-convex fluxes and fourth order diffusion have stable undercompressive fronts, yielding such unusual behavior as double shock structures from simple jump (Riemann) initial data. Thermal/gravity driven thin film flow is described by such equations and the signature of undercompressive fronts has been observed in recent experiments. Unlike compressive fronts, undercompressive film fronts are stable to fingering instabilities.