VISCOUS CAHN-HILLIARD EQUATION .2. ANALYSIS

The viscous Cahn-Hilliard equation may be viewed as a singular limit o f the phase-held equations for phase transitions. It contains both the Allen-Cahn and Cahn-Hilliard models of phase separation as particular cases; by specific choices of parameters it may be formulated as a on e-parameter (say alpha) homotopy connecting the Cahn-Hilliard (alpha=0 ) and Allen-Cahn (alpha=1) models. The limit alpha=0 is singular in th e sense that the smoothing property of the analytic semigroup changes from being of the type associated with second order operators to the t ype associated with fourth order operators. The properties of the grad ient dynamical system generated by the viscous Cahn-Hilliard equation are studied as alpha varies in [0, 1]. Continuity of the phase portrai ts near equilibria is established independently of alpha is an element of [0, 1] and, using this, a piecewise, uniform in time, perturbation result is proved for trajectories. Finally the continuity of the attr actor is established and, in one dimension, the existence and continui ty of inertial manifolds shown and the flow on the attractor detailed. (C) 1996 Academic Press, Inc.