Interface dynamics for an Allen-Cahn-type equation governing a matrix-valued field

Abstract

We consider the initial value problem for the generalized Allen-Cahn equation, \[\partial_t Φ= ΔΦ-\varepsilon^{-2} Φ(Φ^t Φ- I), \qquad x \in Ω, \ t\geq 0,\] where $Φ$ is an $n\times n$ real matrix-valued field, $Ω$ is a two-dimensional square with periodic boundary conditions, and $\varepsilon > 0$. This equation is the gradient flow for the energy, $E(Φ) := \int \frac{1}{2} \|\nabla Φ\|^2_F+\frac{1}{4 \varepsilon^2} \| Φ^t Φ- I \|^2_F$, where $\| \cdot \|_F$ 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 single-signed determinant, at each point of the domain, at a fast $O(\varepsilon^{-2} t)$ 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 $O(\varepsilon t)$ 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 $n\geq 2$ 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.