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

## TL;DR

This paper analyzes the dynamics of a matrix-valued Allen-Cahn equation, revealing how solutions evolve towards orthogonal matrices and how interfaces move under curvature and surface diffusion effects.

## Contribution

The paper introduces asymptotic methods to describe the solution behavior of a matrix-valued Allen-Cahn equation, including interface evolution driven by curvature and surface diffusion.

## Key findings

- Solutions rapidly approach orthogonal matrices for single-signed determinant initial conditions.
- Interface evolution is driven by curvature and surface diffusion, depending on matrix dimension.
- Numerical experiments support the analytical asymptotic descriptions.

## Abstract

We consider the initial value problem for the generalized Allen-Cahn equation, \[\partial_t \Phi = \Delta \Phi-\varepsilon^{-2} \Phi (\Phi^t \Phi - I), \qquad x \in \Omega, \ t\geq 0,\] where $\Phi$ is an $n\times n$ real matrix-valued field, $\Omega$ is a two-dimensional square with periodic boundary conditions, and $\varepsilon > 0$. This equation is the gradient flow for the energy, $E(\Phi) := \int \frac{1}{2} \|\nabla \Phi \|^2_F+\frac{1}{4 \varepsilon^2} \| \Phi^t \Phi - I \|^2_F$, 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.