ALEX 2018 Workshop: Abstracts

Evolution problems for interfaces: Solution concepts and their uniqueness properties

Julian Fischer and Sebastian Hensel

Institute for Science and Technology Austria, Klosterneuburg (Austria)

For many evolution problems for interfaces – like for example the free boundary problem for the Navier-Stokes equation for two immiscible fluids or mean curvature flow – varifold solutions are known to exist globally in time, but the uniqueness of varifold solutions is either unknown or even known to fail in general. At the same time, strong solution concepts are in general limited to local in time existence results due to the development of geometric singularities. In the absence of a comparison principle, the relation between varifold solutions and strong solutions for interfacial evolution problems has remained a mostly open question. We describe a concept of relative entropies for interfacial evolution problems, which enables us to derive a weak-strong uniqueness principle for varifold solutions to the free boundary problem for the Navier-Stokes equation for two immiscible incompressible fluids.