Multiphase Mean Curvature Flow: Uniqueness Properties of Weak Solution Concepts and Phase-Field Approximations

 Topology changes occur naturally in geometric evolution equations like mean curvature flow. As classical solution concepts break down at such geometric singularities, the use of weak solution concepts becomes necessary in order to describe topological changes.
For two-phase mean curvature flow, the theory of viscosity solutions by Chen-Giga-Goto and Evans-Spruck provides a concept of weak solutions with basically optimal existence and uniqueness properties.
In contrast, the uniqueness properties of weak solution concepts for multiphase mean curvature flow had remained mostly unexplored.

By introducing a novel concept of "gradient flow calibrations", we establish a weak-strong uniqueness principle for multiphase mean curvature flow: Weak (BV) solutions to multiphase mean curvature flow are unique as long as a classical solution exists. In particular, in planar multiphase mean curvature flow, weak (BV) solutions are unique prior to the first topological change. As basic counterexamples show, the uniqueness of evolutions may fail past certain topology changes, demonstrating the optimality of our result.

In the last part of the talk, we discuss further applications of our new concept, including the quantitative convergence of diffuse-interface (Allen-Cahn) approximations for multiphase mean curvature flow.