Phase Field Equations with Memory: The Hyperbolic Case

Abstract

We present a phenomenological theory for phase transition dynamics with memory which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential. Thereafter, we focus on the implications of our theory in the hyperbolic case, and we derive asymptotically an equation of motion in two dimensions for the interface between two different phases. This equation can be considered as a hyperbolic generalization of the classical flow by mean curvature equation, as well as a generalization of the Born--Infeld equation. We use a crystalline algorithm to study the motion of closed curves for the generalized hyperbolic flow by mean curvature equation our hyperbolic generalization of flow by mean curvature and present some numerical results which indicate that a certain type of two-dimensional relaxation damped oscillation may occur.