| Oct 28, 2010 |
Carlos Rocha
(Instituto Superior Tecnico, Lisbon, Portugal) |
A permutation characterization of Sturm global attractors of Hamiltonian type |
|
We consider semiflows generated by Neumann boundary value problems of
the form ut = uxx + f(u) on the interval 0 ≤ x ≤ π for
dissipative nonlinearities f = f(u). In the much more general case
f = f(x,u,ux) a permutation characterization for the global attractors
of these semiflows is well known. Here we present a permutation
characterization for the global attractors in the restrictive class of
nonlinearities f = f(u). In this class the stationary solutions of the
parabolic equation satisfy the second order ODE v'' + f(v) = 0 and we
obtain the permutation characterization from a characterization of the
set of 2π-periodic orbits of this planar Hamiltonian system.
This is based on a joint work with Bernold Fiedler and Mathias Wolfrum.
|
| Nov 4, 2010 |
Anna Karnauhova
(Free University Berlin) |
A model of hyperbolic space and the geodesic flow |
Markus Mittnenzweig
(Free University Berlin) |
Bistability of Actin - An Elastic Network Approach |
Hannes Stuke
(Free University Berlin) |
Variational principles of PDEs |
| Nov 11, 2010 |
Johannes Buchner
(Free University Berlin) |
Inhomogeneous Mixmaster Cosmologies |
The BKL conjecture suggests that the approach to the initial singularity
("big bang") is vacuum dominated, local and oscillatory. After some
results have been achieved in an spatially homogeneous setting of
Bianchi class A ("Mixmaster"-models) and in an inhomogeneous, but
non-oscillatory setting of Gowdy-spacetimes, the next step is to
consider inhomogeneous oscillatory cosmological models. The simplest
interesting case is given by the G2-cosmologies.
I will present a version of the evolution equations of G2 models that
directly contain the the Gowdy spacetimes as well as the oscillatory
Bianchi-model of class B. Then I will focus on the latter and explain
the "frame transitions" that appear in the oscillations towards the
singularity. Finally, I show the existence of a heteroclinic 3-cycle and
discuss its properties in the spirit of results that have been achieved
in Bianchi class A.
A particular complication in inhomogeneous models is the occurrence of
spikes, i.e. the formation of spatial structure. In an outlook, I will
discuss how the knowledge of the spatially homogeneous background dynamics
helps to understand the formation of true and false spikes in G2 models.
|
| Nov 25, 2010 |
Johannes Buchner
(Free University Berlin) |
Partially Hyperbolic Fixed Points and Cosmology |
|
In the paper "Partially Hyperbolic Fixed Points" by Floris
Takens, published in "Topology Vol. 10", 1971, a local linearization
theorem is proved for diffeomorphisms and fixed points of flows that are
not hyperbolic but satisfy certain "Sternberg Non-Resonance Conditions".
In my talk, I will review the result and try to give an overview of the
proof. If time permits, I will also discuss the application of the
result to cosmological models of Bianchi type IX (as done by Francois
Béguin) as well as possible future applications of the result to Bianchi
models of class B.
|
| Dec 2, 2010 |
Taras Girnyk
(WIAS Berlin) |
Multistability of twisted states in non-locally coupled Kuramoto-type models |
|
We consider a ring of N Kuramoto oscilators with non-local
repulsive coupling. We obtain sufficient conditions for stability of
so-called twisted states. We discover new types of solutions and
establish a basic framework for their description. We discover
phenomenon of spatial chaos in repulsive networks of Kuramoto oscillators.
|
| Dec 9, 2010 |
Hannes Stuke
(Free University Berlin) |
A variational principle for PDEs |
|
It is well known that a weak solution of certain evolution PDE
could be obtained from the minimizer of some energy functional. In this
talk, I will present a variational principle mostly developed by Nassif
Ghoussoub, which allows the variational resolution for many
non-Euler-Lagrange type PDEs.
|
| Dec 16, 2010 |
Juliette Hell
(Free University Berlin) |
News of the BKL conjecture |
|
I will try to give an overview of the paper by Reiterer and
Trubowitz, "The BKL Conjectures for Spatially Homogeneous Spacetimes".
They prove tumbling universe (for Bianchi VIII and IX, vacuum) for a
more "generic" class than the other results so far (Beguin,
Liebscher et al): they exhibit a set of initial conditions with positive
Lebesgue measure showing this behaviour.
|
| Jan 13, 2011 |
Martin Vaeth
(Free University Berlin) |
Bifurcation for a Reaction-Diffusion System with Pure Neuman Boundary
Conditions and Inequalities |
|
For a reaction-diffusion system which is subject to Turing's effect of
diffusion-driven instability, endowed with Neumann-Signorini boundary
conditions, there is a branch of bifurcations of stationary solutions in
a parameter domain where the same system with only classical Neumann
conditions is stable. In the presence of additional Dirichlet
conditions, this is well known, but for pure Neumann boundary
conditions, such a bifurcation was an open problem for a long time. The
main purpose of the talk is to sketch the proofs, in particular how to
calculate the Leray-Schauder degree for the maps associated to the problem.
|
| Jan 20, 2011 |
Alexander Skubachevskii
(Peoples' Friendship University of Russia) |
The Vlasov-Poisson Equations in Half-Space |
|
The Vlasov-Poisson equations describe the evolution of densities for
electrons and ions in rarefied plasma. Existence and uniqueness of
generalized solutions for such equations in domains with boundary or in
the whole three-dimensional space was studied by many mathematicians. In
this lecture we consider existence and uniqueness of classical solutions
to the Vlasov-Poisson equations in half-space.
|
| Jan 27, 2011 |
Johannes Buchner
(Free University Berlin) |
The Proof of Taken's Linearization Theorem |
|
Taken's linearization theorem provides a local linearization for fixed
points of diffeomorphisms or flows that are not hyperbolic but satisfy
certain Sternberg Non-Resonance Conditions.
It is a generalization of the theorems of Grobman-Hartman and Sternberg,
which provide a linearization near a hyperbolic fixed point.
I will present the proof and focus on the case that is relevant for the
application of the theorem to Bianchi cosmologies, and restrict to that
case (line of equilibria, diagonal linearization, etc) where helpful.
|
| Feb 3, 2011 |
Johannes Buchner
(Free University Berlin) |
The Proof of Taken's Linearization Theorem, Part II |
Last change: Mar. 29, 2011
