Student Harmonic Analysis and PDE Seminar (HADES): Weak-strong uniqueness for multiphase mean curvature flow

Seminar | April 23 | 3:40-5 p.m. | 740 Evans Hall

 Tim Laux, UC Berkeley

 Department of Mathematics

Multiphase mean curvature flow has, due to its importance in materials science, received a lot of attention over the last decades. On the one hand, there is substantial recent progress in the construction of weak solutions. On the other hand, strong solutions are—in particular in the planar case of networks—very well understood.

In this talk, after giving an overview of the topic, I will present a weak-strong uniqueness principle for multiphase mean curvature flow: 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.

In our proof we construct 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.

For the purpose of this talk, I will focus on two instructive model cases: a single smooth interface and a single triple junction.

This is a joint work (in progress) with Julian Fischer, Sebastian Hensel, and Thilo Simon.