Global convergence for Allen–Cahn systems with continuous wells

Establish global-in-time convergence to mean curvature flow for vector-valued Allen–Cahn systems whose zero sets of the potentials have continuous components, including systems with two connected components of the zero set.

Background

The paper proves global, unconditional convergence for a specially designed vector-valued Allen–Cahn system with discrete wells corresponding to finitely many pure phases. The authors contrast this result with systems whose potentials vanish on continuous sets, such as the complex-valued Ginzburg–Landau equation, the Keller–Rubinstein–Sternberg model, and liquid-crystal models.

For these continuous-well systems, short-time convergence results are available in some cases, but the authors state that global-in-time convergence remains unresolved, even when the zero set of the potential has only two connected components. The difficulty is linked to the absence of the codimension-one monotonicity structure used in the paper.