Sharp-interface limit of coupled Allen–Cahn equations

The problem seeks a rigorous unconditional passage from a coupled diffuse-interface evolution to multiphase mean-curvature flow. Earlier results required either a smooth limiting flow or an additional energy-convergence assumption.

Known results

• A 2018 result, based on a 2016 preprint, treats general potentials in arbitrary dimension but assumes convergence of the time-integrated energies.
• A 2022 preprint proves quantitative convergence for suitable $N$-well potentials in dimensions $2$ and $3$, with well-prepared data and a strong limiting flow; it is not expected to extend beyond the first topology change.

August 27, 2026 global convergence claim

The authors of Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow claim the first global unconditional convergence theorem for the coupled system. If correct, it removes the earlier conditional assumptions; the unrefereed preprint remains unverified.

Current status (as of August 2026): A global unconditional theorem is claimed in an unrefereed preprint, while independent verification is absent; the earlier conditional and limited-time results are established.