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

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.

analysis of PDEs

Audience: researchers in the topic

Comments: Here is the poster of this talk: www.dropbox.com/s/807vg0v0nrmlp2d/PDE%20Seminar%207th.pdf?dl=0 Please visit our website to get more information: nguyenquochung1241.wixsite.com/qhung/post/pde-seminar-via-zoom

PDE seminar via Zoom