The sharp interface limit of the matrix-valued Allen-Cahn equation
Xingyu Wang
TL;DR
The paper develops a rigorous sharp interface limit for the matrix-valued Allen–Cahn equation with Saint Venant–Kirchhoff type potential by combining the modulated energy (relative entropy) method with weak convergence techniques. It reformulates the vector-valued dynamics using a Chen–Shatah wedge-product commutator to enable precise energy estimates and constructs a mollified quasi-distance to control the interface profile without high-order asymptotics. Under a mean curvature flow for the interface, the authors prove convergence to a limiting system consisting of harmonic map heat flows in the bulk regions, a mean-curvature driven interface, a minimal-pair condition across the moving interface, and a Neumann-type jump condition across the interface; a weak solution to this limiting system is obtained. The work also provides a well-prepared initial data construction with energy of order $\varepsilon$ and demonstrates uniform bounds that permit passage to the limit $\varepsilon\to 0$, extending the applicability of the modulated energy framework to matrix-valued, partially minimally paired models and offering a robust approach that can be adapted to broader potentials.
Abstract
In this work, we study a matrix-valued Allen-Cahn equation with a Saint Venant-Kirchhoff potential $F(\mathbf{A})=\frac{1}{4}\|\mathbf{A}\mathbf{A}^\top-\mathbf{I}\|^2$. Our approach employs the modulated energy method together with weak convergence methods for nonlinear partial differential equations. This avoids the subtle spectrum analysis of the linearized operator at the so-called quasi-minimal orbits as well as the construction of asymptotic expansion. Moreover, it relaxes the assumption on the admissible initial data, which exhibits a phase transition along an initial interface. As a byproduct, we construct a weak solution to the limiting harmonic heat flow system with both minimal pair and Neumann-type boundary conditions across the interface.