For a vector-valued Allen-Cahn equation with two sphere wells, the ε→0 limit is mean curvature flow coupled to harmonic map heat flow into S^{n-1} with a mixed boundary condition, rigorously justified by matched asymptotic expansions.
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Asymptotic limit of a vector-valued Allen-Cahn equation for phase transition dynamics
For a vector-valued Allen-Cahn equation with two sphere wells, the ε→0 limit is mean curvature flow coupled to harmonic map heat flow into S^{n-1} with a mixed boundary condition, rigorously justified by matched asymptotic expansions.