Interface evolution problems in fluid mechanics and geometry: (Non-)Uniqueness of solutions and weak-strong uniqueness principles

In evolution equations for interfaces, topological changes and geometric singularities often occur naturally, one basic example being the pinchoff of liquid droplets. As a consequence, classical solution concepts for such PDEs are naturally limited to short-time existence results or particular initial configurations like perturbations of a steady state. At the same time, the transition from strong to weak solution concepts for PDEs is prone to incurring unphysical non-uniqueness of solutions. In the absence of a comparison principle, the relation between weak solution concepts and strong solution concepts for interface evolution problems has remained a mostly open question. We establish weak-strong uniqueness principles for two interface evolution problems, namely for planar multiphase mean curvature flow and for the evolution of the free boundary between two viscous fluids: As long as a classical solution to these evolution problems exists, it is also the unique BV solution respectively varifold solution.