Session Cd - Interfacial & Thin-Film Instabilities.
ORAL session, Sunday, November 23
304, Moscone Center
Hocherman and Rosenau (Physica D, 67):113-125, 1993 conjectured that long-wave unstable Cahn-Hilliard type interface models develop finite time singularities when the nonlinearity in the destabilizing term grows faster at larger amplitudes than nonlinear effects in the stabilizing term. We revise their conjecture for a subclass of models, corresponding to a family of equations often used to model thin films in a lubrication context, to say that the destabilizing term can be stronger (by up to a power of two in the nonlinear diffusion coefficient) yet the solution still remains globally bounded. This alternate scaling is motivated by a conservation of volume constraint, and can be rigorously proved for the bounded solutions. We present numerical computations of singularities near the borderline case of this conjecture, confirming both the blowup and global boundedness regimes. We also discuss relationship with recent results (A. J. Bernoff and A. L. Bertozzi, Physica D, 85):375--404, 1995 for a Modified Kuramoto Sivashinsky equation.