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Viscous Cahn–Hilliard Equation II. Analysis

Elliott, C. M. and Stuart, A. M. (1996) Viscous Cahn–Hilliard Equation II. Analysis. Journal of Differential Equations, 128 (2). pp. 387-414. ISSN 0022-0396. doi:10.1006/jdeq.1996.0101.

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The viscous Cahn–Hilliard equation may be viewed as a singular limit of the phase-field 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 one-parameter (sayα) homotopy connecting the Cahn–Hilliard (α=0) and Allen–Cahn (α=1) models. The limitα=0 is singular in the sense that the smoothing property of the analytic semigroup changes from being of the type associated with second order operators to the type associated with fourth order operators. The properties of the gradient dynamical system generated by the viscous Cahn–Hilliard equation are studied asαvaries in [0, 1]. Continuity of the phase portraits near equilibria is established independently ofα∈[0, 1] and, using this, a piecewise, uniform in time, perturbation result is proved for trajectories. Finally the continuity of the attractor is established and, in one dimension, the existence and continuity of inertial manifolds shown and the flow on the attractor detailed.

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Additional Information:© 1996 Academic Press. Received 2 December 1994, Revised 19 April 1995. The authors are very grateful to the referee for an extremely helpful and careful reading of an earlier version of this paper. This work was funded by the Office of Naval Research under Contract N00014-92-J-1876, by the National Science Foundation under Contract DMS-9201727, by the SERC under Grant GR/F85659, and by NATO.
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-92-J-1876
Science and Engineering Research Council (SERC)GR/F85659
North Atlantic Treaty Organization (NATO)UNSPECIFIED
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Andrew StuartJ34
Issue or Number:2
Record Number:CaltechAUTHORS:20170609-125856997
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Official Citation:C.M. Elliott, A.M. Stuart, Viscous Cahn–Hilliard Equation II. Analysis, Journal of Differential Equations, Volume 128, Issue 2, 1 July 1996, Pages 387-414, ISSN 0022-0396, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78061
Deposited By: Ruth Sustaita
Deposited On:09 Jun 2017 21:28
Last Modified:15 Nov 2021 17:36

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