Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Helmut Abels
Universität Regensburg
On a New Diffuse Interface Model for Two-Phase Flows with Different Densities
We present a new diffuse interface for a two-phase flow of two partialy miscible viscous incompressible
fluids with different densities. The model is based on a solenoidal velocity field for the fluid mixture and is
thermodynamically consistent and frame indifferent. We briefly discuss its derivation and a recent result
on existence of weak solutions for singular free energy densities.
This is a joint work with Harald Garcke and Günther Grün (modeling) as well as Daniel Depner and
Harald Garcke (analysis).
Robert Denk
Universtität Konstanz
Maximal Lp-regularity of non-local boundary value problems
Maximal Lp-regularity of operators corresponding to boundary value problems is closely related to
the R-boundedness of the resolvent. In several applications nonlocal operators appear, and one has
to consider classes of operators belonging to the Boutet de Monvel calculus. In the talk we present
some results on the R-boundedness of pseudodifferential operators and parameter-dependent Green
operators. As an application, we consider the Stokes equation in cylindrical domains. The talk is based
on joint results with Jörg Seiler.
Mats Ehrnström
Leibniz Universität Hannover
Steady water waves with multiple critical layers
We construct steady and periodic gravity water waves with multiple critical layers. Those are i) waves
with arbitrarily many critical layers and a single crest in each period, and ii) multimodal waves with
several crests and troughs in each period. The mathematical setting is that of the two-dimensional
Euler equations with a free surface, a flat bed, and otherwise periodic boundary conditions. In the
steady frame, this is an elliptic problem in an unknown domain, and the goal is to find small-amplitude
solutions via bifurcation from a solution curve of rotational running streams with stagnation. Using the
Lyapunov-Schmidt reduction we find solutions in the vicinity of a particular class of eigenvalues, some
of which are not simple. The main novelty lies in the type of vorticity distributions considered, which in
their turn influence the kernel of the linearized problem at the bifurcation points. In particular, the problem
admits two-dimensional bifurcation. The talk is based on joint work with J. Escher, G. Villari and E. Wahlén.
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