Long-wave instabilities and saturation in thin film equations

with A. Bertozzi, CPAM 51(1998)625-661.

Abstract

Hocherman and Rosenau conjectured that long-wave unstable Cahn-Hilliard type interface models develop finite-time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term (Phys. D 67:113-125, 1993). We disprove this conjecture for a class of equations, often used to model thin films in a lubrication context, by showing that in fact the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded. For example, they conjecture that for the long-wave unstable equation

h_t = - (hⁿ h_xxx)_x - (h^m h_x)_x,

m>n leads to blow-up. Using a conservation of volume constraint, we conjecture a different critical exponent for possible singularities of non-negative solutions. We prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions. Specifically, for the above equation, solutions are bounded if m<n+2. The bound is proved using energy methods and is then used to prove the existence of non-negative weak solutions in the sense of distributions. We present preliminary numerical evidence suggesting that m>n+2 can allow blow-up.

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A. B. was supported by an ONR Young Investigator/PECASE award and an Alfred P. Sloan Research Fellowship. M. P. was supported by an NSF postdoctoral fellowship while at the Courant Institute and the Ambrose Monell Foundation while at the Institute for Advanced Study.