Arbeitsgemeinschaft Differentialgleichungen

*Donnerstag, 12:00 -14:00 Uhr, Raum 110*

 

Prof. Dr. P. E. Kloeden
Prof. Dr. T. Weth
Prof. Dr. J. Baumeister
aProf. Dr. H. Crauel
Prof. Dr. J. Bliedtner

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Aktuelle Vorträge

 

Jan-Erik Stecher, Universität Heidelberg, 9. Februar 2012

 

Titel: Dirac Concentrations in Lotka-Volterra parabolic PDE

Abstract: In this presentation a diploma thesis based on a paper written by Benoit Perthame and Guy Barles will be discussed.
It deals with a parabolic nonlinear PDE which describes the evolution of species, including small mutations.
We will illustrate existence and uniqueness results and present convergence results by letting mutations vanish.
This leads to a specific structure of the solution. It can be written as a sum of Dirac measures, which can be interpreted as a
mathematical way to support Darwin's Law of Evolution.

 

 

 

 

 

Dr. Georg Schöchtel, Fachbereich Mathematik, TU Darmstadt, 26. Januar 2012

Title: Motion of inertial particles in Gaussian fields driven by an infinite-dimensional fractional Brownian motion

Abstract: The motion of an inertial particle in a fractional Gaussian random field is studied.
The motion is described by Newton's second law for a particle on a 2D torus, with force
proportional to the difference between a background fluid velocity and the particle velocity itself.
The fluid velocity satisfies a linear stochastic PDE driven by an infinite-dimensional fractional
Brownian motion. The usefulness of such random velocity fields in simulations is that we can
generate random velocity fields with a given energy spectrum, thus creating caricatures of realistic
turbulent flows. This model captures also the clustering phenomenon of preferential concentration
observed in real world and numerical experiments, i.e. particles cluster in regions of low vorticity and
high strain rate. We prove almost sure existence and uniqueness of particle paths and give sufficient
conditions to rewrite this system as a random dynamical system with a global random pullback
attractor. Further we give upper bounds of the almost sure constant Hausdorff dimension of the
random attractor. Finally numerical investigations are considered.

 

Prof. Hamid Zangeneh, Isfahan University of Technology,
              z.Zt. Goethe-Universität, 19. Januar 2012

Titel: "Pattern Selection on the Growing Tips of Plants:
             Bifurcations on Spherical Cap"

 

Abstract

 

 

 

 

Nils Ackermann, Universidad Nacional Autonoma de Mexico, 7. Juli 2011

Titel:  Multi-soliton standing waves in expanding waveguides

Abstract: Consider a compact submanifold M without boundary of
N-dimensional Euclidean space and, for R>0 large enough, the tubular
neighborhood U_R of radius 1 of the expanded manifold RM.  Also
consider the time-independent Schrödinger equation with a cubic power
nonlinearity on U_R and Dirichlet boundary conditions.  Solutions to
this equation yield standing light waves for the waveguide U_R, filled
with a self-focusing medium.  We prove, for any natural number n and R
large enough, the existence of a positive solution with n bumps.  If M
is one dimensional, i.e. U_R is a tubular guide, we may allow M to
have boundary and prove existence of multi-bump solutions with
alternating signs along the tube.  The proof rests on Lyapunov-Schmidt
reduction to the configuration space of M and on asymptotic estimates
of the variational functional.

 

 

 

Maria Anguiano, University of Sevilla, 14. April 2011

Titel: Existence of pullback attractors for nonlinear and nonautonomous parabolic PDEs

Abstract: "The understanding of the asymptotic behaviour of dynamical systems
is one of the most important problems of modern mathematical physics.
One way to treat this problem for systems having some dissipativity properties
is to analyze the existence and structure of its global attractor.
On some occasions, some phenomena are modelled by nonlinear evolutionary
equations which do not take into account all the relevant information of the real
systems. Instead some neglected quantities can be modelled as an external force
which in general becomes time-dependent. For this reason, non-autonomous
systems are of great importance and interest. In this talk I will analyze several
models of reaction-diffusion equations in some bounded and unbounded domains.
I will use the pullback theory so much for single-valued as for multi-valued non-
autonomous dynamical systems, since this allows for more generality in the
non-autonomous terms, to prove the existence of pullback attractors for our models
of reaction-diffusion equations."

Juan Carlos de los Reyes, TU Berlin, 17. Februar 2011

Titel: On semi-smooth Newton methods for the numerical solution of viscoplastic
fluid flow and related variational inequalities

Abstract: In this talk we focus on numerical optimization techniques for the solution
of Bingham viscoplastic fluid flow and other related variational inequalities. Bingham
materials are characterized by the presence of a so-called yield stress: they behave like
solids in regions where the stresses are small and like incompressible fluids where
the stresses are larger than a plasticity threshold. Based on both primal and dual formulations
of the problem, two regularization strategies will be presented. The well-posedness of
each regularized problem is verified and convergence of the regularized solution towards
the original one is studied.
For the solution of each regularized system, generalized Newton algorithms are constructed.
We will present results on global and local superlinear convergence in a function space
setting and/or in finite dimensional spaces, after discretization of the systems. Also
continuation strategies, based on the properties of the path-value functional, are designed and numerically tested.
Finally, since a similar yield behavior is also found in problems arising from other application
areas like elastic contact, plasticity or image denoising, the applicability of the approach
to other variational inequalities will be discussed.

 

 

Univ.Prof.Dr. Ronny Ramlau (Universität Linz), 10. Februar 2011

Titel: tba

 

Dr. Matthias Kurzke (Universität Bonn),  13. Januar 2011

Titel: Vortex motion in the Landau-Lifshitz-Gilbert equations

Abstract: The Landau-Lifshitz-Gilbert equations (modeling the evolution of ferromagnets) are a hybrid of harmonic map heat flow and the Schrödinger map flow. In certain geometries, vortex-like singularities are naturally observed. We derive the equation of motion for these singularities.

(This is joint work with C. Melcher, R. Moser, D. Spirn.)

 

 

 

Archiv

 

Denis Bonheure (Universite Libre de Bruxelles), 10. Juni 2010

Symmetry and multiplicity of least energy solutions for semilinear elliptic boundary value problems.

Abstract : We review some well-known as well as recent results about symmetry, uniqueness or multiplicity of positive or nodal solutions of homogeneous Dirichlet boundary value problems. We then discuss analogous problems with Neumann boundary conditions and we emphasize the drastic differences and open questions.

Dr. Arno Berger (University of Alberta), 20. Mai 2010

Dynamics in finite time - Some thoughts on concepts and applications

Abstract: Motivated strongly by applications, notably in geophysical
fluid dynamics, finite-time dynamics aims at identifying and systematically studying dynamical roperties of systems that are defined only over a bounded interval of time. Many classical symptotic concepts do not apply in this situation and have to be modified or replaced altogether. Quite a number of modified or new concepts have recently been proposed in this regard, and finite-time dynamics has become an active and diverse field. Much attention is being focused on the development of finite-time stability, spectral and bifurcation theories that are both practicable and consistent with their classical counterparts. A key ingredient in these (and other) areas is hyperbolicity which, ever since the dawning of dynamical systems theory, has been recognised as a fundamental concept. Not too surprisingly, when reformulated appropriately hyperbolicity emerges as a key notion in the finite-time context as well.

After providing a brief overview of finite-time dynamics in general, and hyperbolicity in the classical and the finite-time settings in particular, this talk will motivate and present several results that generalise and unify earlier work. Specifically, the existence, non-uniqueness and robustness of finite-time (un)stable manifolds will be discussed, as well as the basic problem of detecting  Lagrangian) hyperbolicity from (Eulerian) data encoded in a dynamic partition of the extended phase space. Some of the fundamental challenges inherent to finite-time dynamics, both practical and conceptual, will become apparent.

 

Prof. Ken Palmer (National University of Taiwan, z. Zt Universitaet Tuebingen), 6. Mai 2010

Lyapunov Exponents and Sensitive Dependence

In general if an orbit of a one-dimensional map has positive Lyapunov exponent, it need not exhibit sensitive dependence. However if the map is smooth and the orbit stays away from critical points, then positive Lyapunov exponent does imply sensitive dependence. Also we exhibit a class of maps for which positive Lyapunov exponent does imply sensitive dependence even for orbits which have critical points in their omega limit set.

 

Hugo Tavares (University of Lisbon, z.Z. Frankfurt), 22. April 2010

Asymptotic study of a reaction diffusion system with competition terms

Abstract: We take a system of partial differential equations with competition terms which arises in the phenomenon of Bose-Einstein condensation. For such system, we study the asymptotics of its solutions as the competition term goes to infinity. The limiting profiles are segregated, and we study the properties of their nodal sets. We present a general regularity theory which can be applied in other situations.

(joint works with B. Noris, G. Verzini and S. Terracini)

 

Martin Riedler (Heriot Watt University, Edinburg), 11. Februar 2010

A stochastic model for voltage potential transport in passive neuronal membranes

Abstract: In my talk I will give a short introduction to the typical approaches of modelling inneuroscience and characteristic problems and questions that arise. Particularly, I will focus on hybrid stochastic models which represent the class of models for single neurons or neuronal membranes, respectively, closest to the biophysical realityand still analytically tractable and practically relevant. To this end I will introduce aa class of models employing Piecewise Deterministic Processes which give a closedand mathematically precise description of a stochastic neuronal membranes based onfundamental and biophysical accurate premises.
Further, I will present algorithms for the (approximate) simulation of the neuron models we are considering. I will also present error estimates and sketch the proof of convergence for the algorithms.

 

Christophe Troestler (Université de Mons-Hainaut, Belgien), 28. Januar 2010

Oddness of least energy nodal solutions of the Lane-Emden problem.

Abstract: We will discuss the symmetries of least energy, sign changing, solutions of Lane-Emden type of problems.  In particular, we will be interested in the oddness of these solutions on spherically symmetric domains.  We will outline generalizations to more general domains and non-linearities.

 

Dr. Gilles Evequoz (Goethe-Universität Frankfurt), 10. Dezember2009

Gap solitons for the discrete nonlinear Schrödinger equation with an interface


Abstract: In this talk, we shall discuss a special case of the discrete nonlinear Schrödinger equation on the whole space where the potential and the nonlinearity are both periodic on each side of a given interface, but not globally periodic. We study the existence of ground-state solutions (i.e. solutions having the least energy among all) of the corresponding stationary problem, using a variational method. This is joint work with Prof. Wolfgang Reichel (Universität Karlsruhe)

 

Prof. Dr. Andreas Rieder (Universität Karlsruhe) 19. November 2009

Ein Newton-Löser für die Elektrische Impedanztomographie

 

Abstract: Wir präsentieren ein inexaktes Newton-Verfahren zur effizienten Lösung des inversen Problems der zweidimensionalen Elektrischen Impedanztomographie, wobei wir das sogenannte 'vollständige Elektrodenmodell' zugrunde legen. Im Fokus des Vortrags stehen die Herleitung des Verfahrens, seine Konvergenz und numerische Effizienz. Letztere verdeutlichen wir durch  numerische Resultate.

 

PD Dr. Thomas Lorenz (Goethe-Universität Frankfurt) 29. Oktober 2009

Mutational analyis: for evolutions, think beyond vector spaces

 

Prof. Dr. Oliver Dorn (University of Manchester), 24. September 2009

"A level set approach for structural inversion from indirect data"

Abstract: In many real world applications of science and engineering the task arises to invert for material profiles inside a given domain from indirectly obtained data. These situations occur for example in medical imaging, remote sensing, geophysical tomography or nondestructive testing of materials. The regions of interest are probed in these applications by some kind of fields which propagate inside these regions according to a given physical law, which typically is described by a partial differential equation (PDE). The inverse problem consists of finding a map of the local distribution of the parameters entering in the PDE from data measured outside of the domain of interest. This inverse problem is almost always ill-posed, and stable solutions are only available by adding prior information to the problem. A classical way of doing so is to use so-called Tikhonov Philips functionals, yielding smooth representations of the solutions. When it is known that there are interfaces in the domain, then these classical Tikhonov Philips solutions are suboptimal, and alternative techniques need to be developed. We will present in this talk a novel technique for structural inversion which uses a level set technique for inverting for parameter profiles with interfaces. We demonstrate this quite novel technology for two important applications, namely the early detection of breast cancer using microwaves, and for a crack-detection problem. Numerical simulations in 2D are presented for these two applications which show that the level set technique shows great promise as an alternative tool to classical Tikhonov Philips inversion schemes.

 

Prof. Dr. J. Zubelli (IMPA, Rio de Janeiro), 6. August 200914 Uhr c.t., 711 gr.

"Inverse Problems in Finances: A Short Survey of Calibration Methods"

Abstract: We survey the problem of calibrating the volatility of securities and asset prices by using
quoted prices of financial instruments, such as derivatives. This is done within the framework
of regularization provided by inverse problem theory and leads to very interesting mathematical
and computational problems.We briefly review a number of recent contributions to the field and
highlight its activity.

 

Patrick Grüning (Goethe-Universität Frankfurt)

18. Juni 2009, 14:15 uhr, R. 711gr.

"Multilevel Monte Carlo Simulationen von Optionspreisen und Sensitivitäten des Underlyings"

 

Sven Jarohs (Goethe-Universität Frankfurt)

28. Mai 2009, 14.15 Uhr, R. 711gr.

"Hausdorffmaß von Mengen unter Linearen Operatoren"

 

Isabella Ianni (Sissa, Trieste)

    Thursday, 14 May, 14.15 in room 711 gr.

"Semiclassical standing waves for the nonlinear Schrodinger-Maxwell equation"

Abstract:  We study the semiclassical limit for a nonlinear stationary
Schrodinger-Maxwell system of equations in R3.
This system has been introduced as a model which describes standing waves
for the nonlinear Schrodinger equation interacting with the electrostatic
field.
By using a Lyapunov-Schmidt reduction in a variational setting we prove
the existence of families of solutions which exhibit a concentration behavior.
Precisely we find solutions concentrating pointwise and also radially
symmetric solutions exhibiting concentration around a sphere.
Some stability results are also discussed.

 

Arnulf Jentzen (Doktorand bei Prof. Dr. P. E. Kloeden)

    Tuesday, 5 May 2009, 16.15 in room 110

"Taylor expansions for stochastic partial differential equations"

 

Dr. Gilles Evequoz (Universität Karlsruhe)

    Thursday, 30 April 2009, 14.15 in room 711 gr.

"Hadamard differentiability and bifurcation for some nonlinear elliptic equations"

 

Mouhamed Moustapha Fall (SISSA, Trieste)

        Thursday, 23 April 2009, 14.15 in room 711 gr.

"The Free Boundary Plateau Problem for large H-surfaces"

 

Dr. Herbert Egger (RWTH Aachen)

       Tuesday, 7 April 2009, 14.15 in room 110

 "An Inverse Problem in Computational Finance: Analysis, Regularization and Numerical Solution"

 

 

 

 

 

 

 

 

geändert am 06. Februar 2012  E-Mail: Bürobuero@math.uni-frankfurt.de

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