PDE Seminar| Institute of Mathematical Sciences
Time：Thursday, 23:00-23:50 May 21st, 2020
Location：Zoom
Speaker: Julian Fischer,Institute of Science and Technology Austria (IST Austria)
Abstract: In evolution equations for interfaces, topological changes and geometric singularities 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 important 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. In the case of multiphase mean curvature flow, our construction leads to a gradient-flow analogue of the notion of calibrations. Based on joint works with Sebastian Hensel, Tim Laux, and Thilo Simon.
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