Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

Felix Schulze
Freie Universität Berlin
Stability results for curvature flows
DMV Tagung 2011 - Köln, 19. - 22. September
In the first part of this talk we will present several results, which are joint work with M. Simon and O.
Schnürer, concerning long-time behaviour and convergence of the Ricci flow close to the Euclidean and
hyperbolic space. The Euclidean and hyperbolic space are examples of so-called ’self-similar’ solutions
of the Ricci-flow where the geometric shape only changes by scaling.
In the second part we consider the flow of regular networks in the plane under curve shortening flow.
We will explain how a similar stability result - in this case for self-similarly expanding solutions - can be
applied to prove short-time existence from non-regular initial data. This is joint work with A. Neves and T.
Ilmanen.
Tim Seger
Universität Konstanz
Regularity Theory for an Elliptic Parabolic System
In the modeling of Lithium Ion Batteries and Fuel Cells there appears a nonlinear system of one parabolic
and two elliptic equations in a bounded domain Ω that describe the evolution of concentration ce and of
the electric potentials φe and φs in liquid and solid phase respectively. These equations are coupled by a
kinetic expression, that takes the form
S(φs − φe,ce) = c −1/2
e exp(φs − φe) − c 3/2
e exp(φe − φs), (1)
and appears as the right hand side of each of the equations.
Local existence of bounded weak solutions to this system was proved in 2006 by J. Wu, J. Xu and H. Zou.
Under slightly stronger hypothesis on the coefficients we prove the local existence of strong solutions in
the space W 1 p (0,T ;Lp(Ω)) ∩ Lp(0,T ;W 2 p (Ω)) for a smooth domain Ω, using results on elliptic regularity
and Leray-Schauder theory.
Miles Simon
Universität Magdeburg
Ricci-Fluss von Kegeln mit nichtnegativem Krümmungsoperator
Eine Zusammenarbeit mit Felix Schulze. Es sei C ein Kegel mit nichtnegativem Krümmungs-operator,
positiver asymptotischer Volumen-Dichte (asymptotic volume ratio) und beschränkter Krümmung. Wir
zeigen, dass eine globale Lösung zur Ricci-Fluss Gleichung mit Anfangswert C existiert und, dass die
Lösung ein expandierendes Soliton (expanding soliton) ist.
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