Minisymposium “Interfaces and Free Boundaries”
Organizer: Prof. Dr. Julian Fischer
(Tuesday, 03.03.20, 15:30-17:30, lecture hall H12)
Prof. Dr. Harald Garcke (University of Regensburg):
“Free Boundary Problems for the Dynamics of Fluidic Biomembranes”
Biomembranes consisting of multiple phases can undergo complex shape transitions. We study a Cahn-Hilliard model on an evolving hypersurface coupled to (Navier-)Stokes equations on the surface and in the surrounding medium to model these phenomena. The evolution is driven by a curvature energy, modelling the elasticity of the membrane, and by a Cahn-Hilliard type energy, modelling line energy effects.
The model is carefully introduced and analytical properties are shown. A stable finite element approximation is introduced and several complex phenomena occurring for two-phase membranes are computed.
Prof. Dr. Tim Laux (University of Bonn):
“Convergence of the thresholding scheme for multiphase mean curvature flow”
The thresholding scheme is an efficient time discretization for mean curvature flow. Esedoglu and Otto showed that the scheme respects the gradient-flow structure of (multiphase) mean curvature flow as it may be viewed as minimizing movements for an energy that Gamma-converges to the total interfacial area. In this talk I will present some convergence results, in particular in the multiphase case with arbitrary surface tensions which establish the convergence to weak formulations of (multiphase) mean curvature flow in the BV-framework of sets of finite perimeter.
This is joint work with Felix Otto (Max Planck Institute for Mathematics in the Sciences).
Sebastian Hensel (IST Austria):
“Weak-strong uniqueness and stability of evolutions for multi-phase mean curvature flow”
For interface evolution problems not admitting a geometric comparison principle, as it is for instance the case in multi-phase mean curvature flow or fluid-fluid interface evolution, the derivation of a stability estimate or a weak-strong uniqueness principle often represents an open problem. In this talk, I will introduce a notion of gradient flow calibrations which adapts the classical notion for minimizers of the interface area functional from the static setting to the evolutionary setting. This construction in turn serves as the key building block for a multi-phase generalization of the notion of relative entropies for interface evolution problems. Both concepts taken together enable us to establish a weak-strong uniqueness principle for multi-phase mean curvature flow in the plane: We show that as long as a classical solution exists, any weak solution in the sense of the BV formulation of Laux and Otto (Calc. Var. Partial Differential Equations 2016) starting from the same initial data must coincide with it.
This is a joint work with Julian Fischer, Tim Laux and Thilo Simon.
Prof. Dr. Patrick Dondl (University of Freiburg):
“Pinning and depinning of interfaces in random media”
We consider the evolution of an interface, modeled by a parabolic equation, in a random environment. The randomness is given by a distribution of obstacles of random strength. To provide a barrier for the moving interface, we construct a positive, steady state supersolution. This construction depends on the existence, after rescaling, of a Lipschitz hypersurface separating the domain into a top and a bottom part, consisting of boxes that contain at least one obstacle of sufficient strength. Furthermore, we examine the question of existence of a solution propagating with positive velocity in a random field with non-bounded random obstacle strength. This work shows the emergence of a rate independent hysteresis in systems subject to a viscous microscopic evolution law through the interaction with a random environment.