Speaker
Description
In this talk, I would like to introduce a novel weak solution concept for two-phase Volume Preserving Mean Curvature Flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions consist in evolving varifolds coupled with the phase volumes by a transport equation. We preliminarily concentrate on the existence, showing first that any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to this new definition. Then, we show, for the first time for a minimizing movements scheme, the unconditional convergence towards such a De Giorgi varifold solution, providing an alternative proxy for the completely degenerate $L^2$ distance. Finally, we introduce a new notion of volume-preserving gradient-flow calibrations to show that any classical solution to Volume Preserving Mean Curvature Flow is unique in the class of our new varifold solutions.