On the gradient-flow structure of multiphase mean curvature flow

On the gradient-flow structure of multiphase mean curvature flow - Tim Laux

Tim Laux
University of California, Berkeley
December 9, 2019

Due to its importance in materials science where it models the slow relaxation of grain boundaries, multiphase mean curvature flow has received a lot of attention over the last decades. 

In this talk, I want to present two theorems. The first one is a convergence result for the vector-valued Allen-Cahn equation. Similar to an intriguing proof by  Luckhaus-Sturzenhecker and a recent work with Felix Otto, we derive the convergence of the phase field to a distributional solution of multiphase mean curvature flow. The second result is a weak-strong uniqueness principle: as long as a strong solution to multiphase mean curvature flow exists, any distributional solution with optimal energy dissipation rate has to coincide with this solution. 

Both proofs are based on the gradient-flow structure of multiphase mean curvature flow. In particular, for the first result, we construct a phase-field version of the tilt-excess, a well-known functional in geometric measure theory. For the second result, we define a suitable relative entropy functional, which in this geometric context may be viewed as a time-dependent variant of calibrations. Just like the existence of a calibration guarantees that one has found a global minimum, the existence of a time-dependent calibration ensures that the route of steepest descent in the energy landscape is unique and stable. 

This is joint work with Julian Fischer, Sebastian Hensel, and Thilo Simon.