We will talk about the asymptotical behaviour of a matrix-valued Allen-Cahn equation when a small parameter tends to zero. Precisely, we show that the limit system is a two-phases flow system: the phase interface evolves according to the mean curvature flow; in two bulk phase regions, the solution obeys the harmonic map heat flow into two different manifolds; on the interface, the phase matrices in two sides satisfy a novel mixed boundary condition.
The proof follows the roadmap developed by de Mottoni-Schatzman and Alikakos-Bates-Chen: we first construct an approximate solution solving the regularized system up to arbitrary small terms, and then we prove a spectral lower bound for the linearized operator around it, and finally we estimate the difference between the true solution and the approximate solution. The main difficulties come from the inherent (partially minimal pairing) property of this problem.
This is joint work with Mingwen Fei, Fanghua Lin and Zhifei Zhang.