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**Viscous Cahn-Hilliard equation. II: Analysis.**
*(English)*
Zbl 0855.35067

Summary: [For Part I, cf. Nonlinearity 8, No. 2, 131-160 (1995; Zbl 0818.35045).]

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 (say \(\alpha\)) homotopy connecting the Cahn-Hilliard \((\alpha= 0)\) and Allen-Cahn \((\alpha= 1)\) models. The limit \(\alpha= 0\) 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 as \(\alpha\) varies in \([0, 1]\). Continuity of the phase portraits near equilibria is established independently of \(\alpha\in [0, 1]\) 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.

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 (say \(\alpha\)) homotopy connecting the Cahn-Hilliard \((\alpha= 0)\) and Allen-Cahn \((\alpha= 1)\) models. The limit \(\alpha= 0\) 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 as \(\alpha\) varies in \([0, 1]\). Continuity of the phase portraits near equilibria is established independently of \(\alpha\in [0, 1]\) 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.

### MSC:

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

47D06 | One-parameter semigroups and linear evolution equations |