TY - JOUR
T1 - Energy levels of steady states for thin-film-type equations
AU - Laugesen, R. S.
AU - Pugh, M. C.
N1 - 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.
Laugesen was partially supported by NSF Grant Number DMS-9970228, by a grant from the University of Illinois Research Board, and by a fellowship from the University of Illinois Center for Advanced Study. He is grateful for the hospitality of the Department of Mathematics at Washington University in St. Louis.
PY - 2002/7/1
Y1 - 2002/7/1
N2 - We study the phase space of the evolution equation ht = -(f(h)hxxx)x - (g(h)hx)x by means of a dissipated energy. 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 ∞, or tend to a nonzero constant. We first show that all positive periodic steady-states are saddles in the energy landscape, with respect to zero-mean perturbations, 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) = yn and g(y) = ℬym for some ℬ > 0) we analytically determine the relative energy levels of distinct steady-states. For example, with m - n ∈ [1, 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 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 for the energy. 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.
AB - We study the phase space of the evolution equation ht = -(f(h)hxxx)x - (g(h)hx)x by means of a dissipated energy. 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 ∞, or tend to a nonzero constant. We first show that all positive periodic steady-states are saddles in the energy landscape, with respect to zero-mean perturbations, 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) = yn and g(y) = ℬym for some ℬ > 0) we analytically determine the relative energy levels of distinct steady-states. For example, with m - n ∈ [1, 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 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 for the energy. 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.
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U2 - 10.1006/jdeq.2001.4108
DO - 10.1006/jdeq.2001.4108
M3 - Article
AN - SCOPUS:0036647446
SN - 0022-0396
VL - 182
SP - 377
EP - 415
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 2
ER -