January  2001, 7(1): 127-146. doi: 10.3934/dcds.2001.7.127

Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations

1. 

Department of Mathematics, Brigham Young University, Provo, UT 84602, United States

Received  August 1999 Revised  November 2000 Published  November 2000

The discrete Allen-Cahn and Cahn-Hilliard equations are (continuous time) lattice differential equations that are analogues of two well-studied parabolic partial differential equations of materials science, and that model the evolution of, respectively, a conserved, or nonconserved, quantity, which has two preferred homogeneous phases and which avoids spatial inhomogenieties. If the interaction length $\nu$ is small, equilibrium states are scattered throughout phase space, but as $\nu$ increases most of these equilibria disappear. As they do so, the system passes through a transitional stage known as dynamical metastability, during which true equilibrium solutions are difficult to distinguish from solutions that evolve slowly but eventually traverse long distances. This paper contains results on the grain sizes of genuine equilibria that help make this distinction possible. In particular, lower bounds on grain sizes and connections between the grain sizes of adjacent grains and next-nearest neighbors are established.
Citation: Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127
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