Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Bogdan Matioc
Leibniz Universität Hannover
On two-phase flows modelling thin films in porous media
Starting from a moving boundary problem which describes the motion of two fluids in a porous medium,
known as the Muskat problem, we pass to the limit of small layer thickness and obtain a second order degenerate
parabolic system of equations. This strongly coupled system is the generalisation of the Porous
Medium equation to the case of two fluids.
We show that if the initial data are nonnegative, then the degenerate system possesses a global solution
which remains nonnegative and which converges exponentially fast towards some flat equilibrium.
This is a joint work with J. Escher and Ph. Laurencot.
Mark Peletier
Eindhoven University of Technology
Passing to the limit in the Wasserstein Gradient-flow formulation
The Wasserstein gradient-flow structure describes a large number of parabolic, diffusive systems. This
structure has been used to derive many properties of such systems, such as well-posedness, stability,
and large-time behaviour. Here we focus on the use of the gradient-flow structure to prove convergence.
Extending ideas of Stefanelli and Serfaty, we use the Wasserstein gradient-flow structure to prove convergence
in a singularly perturbed diffusion problem. Our test problem arises from the interpretation of
chemical reactions as diffusion in a potential landscape, initiated by Wigner and Kramers in the 1930’s. In
this interpretation a reaction event corresponds to the escape of the diffusing particle from one potential
well into another. In earlier work (with Savare and Veneroni) we proved the convergence of this system
in the limit of high activation energy to the corresponding reaction-diffusion system — but without making
use of the Wasserstein gradient-flow structure.
In this talk I revisit the result, and reprove it using the Wasserstein gradient-flow structure. The method
has some interesting aspects, such as relatively weak compactness requirements, a somewhat surprising
limit, and a tight connection to stochastic particle systems. In addition it has the potential for wide
applicability among the broad class of Wasserstein gradient flows.
This is work with Steffen Arnrich, Alexander Mielke, Giuseppe Savare, and Marco Veneroni.
Wolfgang Reichel
KIT - Karlsruhe Institute of Technology
Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach
spaces
We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on
bounded symmetric domains of R n . The solutions take values in ordered Banach spaces E, e.g. E = R N
ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering
cones which are different from the standard cone of componentwise nonnegative elements we can prove
symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use
methods based on maximum principles (the method of moving planes) suitably adapted to cover the
case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.(With Gerd Herzog,
Karlsruhe.)
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