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

  • Julian Fischer (IST Austria)
E1 05 (Leibniz-Saal)


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.

Katja Heid

Jürgen Jost

Max Planck Institute for Mathematics in the Sciences

Felix Otto

Max Planck Institute for Mathematics in the Sciences