数学科学研究所
Insitute of Mathematical Science

PDE Seminar: Weak-strong uniqueness principles for interface evolution problems in fluid mechanics and geometry

PDE Seminar| Institute of Mathematical Sciences

TimeThursday, 23:00-23:50 May 21st, 2020

LocationZoom

 

Speaker: Julian Fischer,Institute of Science and Technology Austria (IST Austria)

AbstractIn 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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