On uniqueness of weak solution concepts for curvature driven interface evolution in geometry and fluid mechanics

Due to topology changes and geometric singularities, the existence theory for interface evolution problems typically relies on weak solution concepts. In the absence of a comparison principle, the question of their uniqueness properties however remained essentially unexplored for a long time. In this talk, I will present weak-strong uniqueness principles for two important interface evolution problems not admitting a comparison principle: i) the flow of two incompressible, viscous and immiscible fluids in the presence of surface tension, and ii) multiphase mean curvature flow. More precisely, for these models we establish uniqueness of a suitable class of weak solutions within the class of classical solutions prior to the first topology change. The key ingredient to these qualitative uniqueness results are quantitative stability estimates in terms of a novel notion of relative entropies for interface energies. I will explain this concept first in a simple two-phase setting. In a second step, I discuss how the corresponding multiphase analogue leads to a gradient flow generalization of a well-known concept from minimal surface theory: the notion of calibrations. I conclude with an outlook on possible future directions. The talk is based on joint works with Julian Fischer, Tim Laux and Theresa Simon.