Energy levels of steady states for thin film type equations

with R. S. Laugesen. Journal of Differential Equations 182(2002)2:377-415

Abstract

We study the phase space of the evolution equation

h_t = -(f(h) h_xxx)_x - (g(h) h_x)_x

by means of a dissipated energy (a Liapunov function). Here h(x,t) >= 0 , and at h=0 the coefficient functions f>0 and g can either degenerate to 0, or blow up to infinity, or tend to a nonzero constant.

We first show all positive periodic steady states are `energy unstable' fixed points for the evolution (meaning the energy decreases under some zero--mean perturbation) if (g/f)'' >= 0 or if the perturbations are allowed to have period longer than that of the steady state.

For power law coefficients (f(y) = y^n and g(y) = B y^m for some B > 0) we analytically determine the relative energy levels of distinct steady states. For example, with 1 <= m-n < 2 and for suitable choices of the period and mean value, we find three fundamentally different steady states. The first is a constant steady state that is nonlinearly stable and is a local minimum of the energy. The second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state; it is a saddle point. The third is a periodic collection of `droplet' (compactly supported) steady states having lower energy than either the positive steady state or the constant one. Since the energy must decrease along every orbit, these results significantly constrain the dynamics of the evolution equation.

Our results suggest that heteroclinic connections could exist between certain of the steady states, for example from the periodic steady state to the droplet one. In a companion article we perform numerical simulations to confirm their existence.

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The authors are grateful to Andrew Bernoff for stimulating comments on the energy landscape of phase space.

Laugesen was partially supported by NSF grant number DMS-9970228, and a grant from the University of Illinois Research Board. He is grateful for the hospitality of the Department of Mathematics at Washington University in St. Louis.

Pugh was partially supported by NSF grant number DMS-9971392, by the MRSEC Program of the NSF under Award Number DMR-9808595, by the ASCI Flash Center at the University of Chicago under DOE contract B341495, and by an Alfred P. Sloan fellowship. Some of the computations were done using a network of workstations paid for by an NSF SCREMS grant, DMS-9872029. Part of the research was conducted while enjoying the hospitality of the Mathematics Department and the James Franck Institute of the University of Chicago.