Interface Dynamics for an Allen--Cahn-Type Equation Governing a Matrix-Valued Field

We consider the initial value problem for the generalized Allen-Cahn equation, partial derivative(t)Phi = Delta Phi - epsilon(-2)Phi(Phi(t)Phi - I), x is an element of Omega, t >= 0, where Phi is an n x n real matrix-valued field, Omega is a two-dimensional square with periodic boundary conditions, and epsilon > 0. This equation is the gradient flow for the energy, E(Phi) := integral 1/2 parallel to del Phi parallel to(2)(F) + 1/4 epsilon(2) parallel to Phi(t)Phi - I parallel to(2)(F), where parallel to . parallel to(F) denotes the Frobenius norm. The primary contribution of this paper is to use asymptotic methods to describe the solution of this initial value problem. If the initial condition has a single-signed determinant, at each point of the domain, at a fast O(epsilon(-2)t) time scale, the solution evolves towards the closest orthogonal matrix. Then, at the O(t) time scale, the solution evolves according to the O-n diffusion equation. Stationary solutions to the O-n diffusion equation are analyzed for n = 2. If the initial condition has regions where the determinant is positive and negative, a free interface develops. Away from the interface, in each region, the matrix-valued field behaves as in the single-signed determinant case. At the O(t) time scale, the interface evolves in the normal direction by curvature. At a slow O(epsilon t) time scale, the interface is driven by curvature and the surface diffusion of the matrix-valued field. For n = 2, the interface is driven by curvature and the jump in the squared tangental derivative of the phase across the interface. In particular, we emphasize that the interface when n >= 2 is driven by surface diffusion, while for n = 1, the original Allen-Cahn equation, the interface is only driven by mean curvature. A variety of numerical experiments are performed to verify, support, and illustrate our analytical results.