### Events (Sommersemester 2019)

Datum Vortragende(r) Titel Raum
14.05.2019
10:30 Uhr
Peter Albers
Universität Heidelberg
Joint Analysis Seminar
Variational methods in generalized billiard dynamics
Hörsaal SeMath, Pontdriesch 14-16

18.06.2019
10:30 Uhr
Bernd Schroers
Heriot-Watt University, Edinburgh
Joint Analysis Seminar
Gauged sigma models and magnetic skyrmions
Hörsaal SeMath, Pontdriesch 14-16

### Vergangene Events

Datum Vortragende(r) Titel Raum
30.04.2019
10:30 Uhr
Denis Serre
ENS de Lyon
Joint Analysis Seminar
Compensated integrability and applications to PDEs and fluid dynamics
Hörsaal SeMath, Pontdriesch 14-16

Positive semi-definite symmetric tensors whose divergence is a bounded measure enjoy a property of enhance integrability. This tool can be used to find new estimates in fluid dynamics, at every level of description, from microscopic to macroscopic. I shall explain who this theory is related to Monge-Ampère equation and transport theory. Then I shall discuss Euler and Boltzmann equations, and molecular dynamics

02.04.2019
10:30 Uhr
Rémy Rodiac
l’Université Catholique de Louvain-la-Neuve
Oberseminar
Two-components Bose-Einstein condensates with spin-orbit coupling
Hörsaal SeMath, Pontdriesch 14-16

In this talk we will study a 1D Gross-Pitaevskii functional modelling the behaviour of a Bose-Einstein condensate with two components interacting via their spin. For some values of the interaction parameter we show that we can reduce the problem to a Modica-Mortola (or Allen-Cahn) type phase transition problem with a non-homogeneous Neumann boundary data. According to the size of this boundary data two situations appear. When the boundary data is small enough then one component fills almost all the domain and the second one surrounds it at the boundary. When the boundary data is big enough, multiple periodic transitions occur and one observe that the two components takes the form of stripes. This is a joint work with A.Aftalion (EHESS).

07.02.2019
10:30 Uhr
Shankar Venkataramani
University of Arizona (USA)
Weekly Seminar
Hyperbolic Monge-Ampere equations and applications to thin sheet elasticity
Hörsaal SeMath, Pontdriesch 14-16

I will talk about some geometric questions that arise in the study of soft/thin objects with negative curvature. After reviewing basic ideas from discrete differential geometry (DDG), I will motivate the need for new "geometric" methods, based on DDG, for studying the mechanics of leaves, flowers, and sea-slugs. I will present some of our results in this direction. This is joint work with Toby Shearman and Ken Yamamoto.

29.01.2019
10:30 Uhr
Paul Sutcliffe
Durham University
Joint Analysis Seminar
Hopfions
Hörsaal SeMath, Pontdriesch 14-16

15.01.2019
10:30 Uhr
Eugene Stepanov
Steklov Mathematical Institute, St. Petersburg
Joint Analysis Seminar
Superposition principles, continuity equations and curves of measures
Hörsaal SeMath, Pontdriesch 14-16

We will consider three different classes of results regarding the structure of curves of measures in metric spaces, all frequently referred to as “superposition principles”, in particular: (A) the results relating the family of measures solving the first order PDE called continuity equation, with the measures on the space of solutions to the associated characteristic ODE; (B) the results giving the structure of curves of positive Borel measures absolutely continuous with respect to some Kantorovich-Wasserstein distance, in terms of measures over some space of rectifiable curves, and, finally, (C) the results on decomposition (or, more appropriately, “disintegration”) of general normal currents (in the sense of De Rham or in the sense of De Giorgi-Ambrosio-Kirchheim) in simpler ones associated with rectifiable curves. We will discuss the modern general results relating (A), (B) and (C) as well as their applications, in particular showing that all those principles descend from a single superposition principle for normal metric currents.

04.12.2018
10:30 Uhr
Peter Hornung
TU Dresden
Joint Analysis Seminar
Extrinsic convexity of W^{2,2} immersions with positive Gauss curvature
Hörsaal SeMath, Pontdriesch 14-16

23.10.2018
10:30 Uhr
Maria J. Esteban
Université Paris-Dauphine
Joint Analysis Seminar
Interpolation inequalities and spectral estimates for Schrödinger magnetic operators
Hörsaal SeMath, Pontdriesch 14-16

In this talk I will present how to prove interpolation inequalities, with as good constants as possible, for Schrödinger operators involving an external magnetic field. These inequalities can be then used to prove spectral estimates for the same kind of operators, but this time involving also an external electrical field.

16.07.2018
13:00 Uhr
Richard Schubert
Bonn
Seminar
Viscosity and conductivity formulas for dilute suspensions
Raum 001, Pontdriesch 14-16

Abstract: Einstein's formula for the effective viscosity of dilute suspensions describes how rigid particles immersed in a Stokes fluid increase its macroscopic viscosity. However, up to now, a rigorous justification has only been obtained for dissipation functionals of the flow field. In this talk, I will discuss ideas for the investigation of many-particle Stokes systems, e.g. the so-called method of reflections and dipole approximations. I will present a rigorous result for the number of particles tending to infinity. Namely, in the limit, the suspension behaves like a fluid with increased viscosity that satisfies Einstein's formula. I will also show how the discussed methods lead to a similar result regarding Maxwell's formula for the conductivity of suspensions.

15.06.2018
14:15 Uhr
Stephanos Venakides
Duke University
Mathematisches Kolloquium
Overview and calculation of dispersive phenomena in integrable systems
Hörsaal V, Hauptgebäude, Templergraben 55

24.04.2018
10:30 Uhr
Enno Lenzmann
University of Basel
Joint Analysis Seminar
Energy-Critical Half-Wave Maps: Solitons and Lax Pair Structure
Hörsaal SeMath, Pontdriesch 14-16

The eneryg-critical half-wave maps equation arises as a classical continuum limit of Calogero-Moser and Haldane-Shastry type spin systems in one space dimension. In this talk, I will discuss some essential features such as the complete classification of traveling solitary waves with finite energy, by using a close connection to minimal surfaces with free and non-free boundary conditions. Furthermore, I will present a recently found Lax pair structure and we explain its potential applications to the dynamics of the half-wave maps equation. Finally, I will mention some open problems. This talk is based on joint work with P. Gérard (Orsay) and A. Schikorra (Pittsburgh).

20.04.2018
14:15 Uhr
Israel Michael Sigal
University of Toronto
Mathematisches Kolloquium
Magnetic vortex lattices
Hörsaal V, Hauptgebäude, Templergraben 55

The Ginzburg - Landau equations play a fundamental role in various areas of physics, from superconductivity to elementary particles. They present the natural and simplest extension of the Laplace equation to line bundles. Their non-abelian generalizations - Yang-Mills-Higgs and Seiberg-Witten equations have applications in geometry and topology. Of a special interest are the least energy (per unit volume) solutions of the Ginzburg - Landau equations. Though the equations are translation invariant, these turned out to have a beautiful structure of (magnetic) vortex lattices discovered by A.A. Abrikosov. (Their discovery was recognized by a Nobel prize. Finite energy excitations are magnetic vortices, called Nielsen-Olesen or Nambu strings, in particle physics.) I will review recent results about the vortex lattice solutions and their relation to the energy minimizing solutions on Riemann surfaces and, if time permits, to the microscopic (BCS) theory.

08.03.2018
10:30 Uhr
Upanshu Sharma
CERMICS, École des Ponts ParisTech
Oberseminar:
Quantifying coarse-graining error
Room 203, Pontdriesch 14-16

Abstract: Coarse-graining or dimension reduction is the procedure of approximating a large and complex system by a simpler and lower-dimensional one. A key feature that allows for such an approximation is a choice to consider only part of information by means of a coarse-graining map F that is strongly many-to-one. Assuming that the configuration of the full system is governed by a stochastic differential equation (SDE), for, say, a random variable X (representing for instance the position of particles in the system), Gyöngy postulated an ‘closed' evolution equation for the reduced (coarse grained) variable F(X), which is again a stochastic differential equation with coefficients derived from the full one. While lower dimensional this evolution is difficult to work with numerically. Legoll and Lelièvre proposed an approximation to this evolution and showed in the case of one-dimensional coarse-graining maps F and starting with reversible SDEs how to estimate the error of this approximation. In this talk, I will present recent generalisations to a more general situations: (1) proving relative entropy estimates using a connection to large deviations and (2) pathwise estimates for non-reversible dynamics.

23.01.2018
10:30 Uhr
Herbert Koch
University of Bonn
Joint Analysis Seminar:
The renormalized nonlinear wave equation in 2d with additive white noise
Hörsaal SeMath, Pontdriesch 14-16

16.01.2018
10:30 Uhr
Rishabh Gvalani
Imperial College London
Oberseminar:
Phase transitions for the McKean-Vlasov equation on the torus
Hörsaal SeMath, Pontdriesch 14-16

We study the McKean-Vlasov equation on the flat torus which is obtained as the mean field limit of a system of interacting diffusion processes enclosed in a periodic box. The system acts as a model for several real world phenomena from statistical physics, opinion dynamics, collective behaviour, stellar dynamics etc.

After commenting on the well-posedness of the equation, we study its long time behaviour and convergence to equilibrium. We then focus our attention on the stationary problem - under certain assumptions on the interaction potential, we show that the system exhibits multiple equilibria which arise from the uniform state through continuous bifurcations. This relates closely with previous work on phase transitions for the Mckean-Vlasov equation (cf. Chayes and Panferov, J. Stat. Phys., 2010). Finally, we attempt to classify continuous and discontinuous transitions for this system and show how this work, in conjunction with previous studies of the system, can be used to recover classical results on phase transitions for the noisy Kuramoto model. This is joint work with José Carrillo, Greg Pavliotis, and André Schlichting.

18.07.2017
14:00 Uhr
Siegfried Beckus
Technion-Israel Institute of Technology
Oberseminar:
Shnol type Theorem for the Agmon ground state
Raum 001, Pontdriesch 14-16

The celebrated Shnol theorem asserts that every polynomially bounded generalized eigenfunction for a given energy E associated with a Schrödinger operator H implies that E is in the L2-spectrum of H. Later Simon rediscovered this result independently and proved additionally that the set of energies admitting a polynomially bounded generalized eigenfunction is dense in the spectrum. A remarkable extension of these results holds also in the Dirichlet setting. It has been conjectured that the polynomial bound on the generalized eigenfunction can be replaced by an object intrinsically defined by H, namely, the Agmon ground state. During the talk, we positively answer the conjecture indicating that the Agmon ground state describes the spectrum of the operator H. Specifically, we show that if u is a generalized eigenfunction for the eigenvalue E that is bounded by the Agmon ground state, then E belongs to the L2-spectrum of H. Furthermore, this assertion extends to the Dirichlet setting whenever a suitable notion of Agmon ground state is available.

18.07.2017
10:30 Uhr
Xian Liao
University of Bonn
Oberseminar:
Conserved energies for the one-dimensional cubic nonlinear Schrödinger equations
Hörsaal SeMath, Pontdriesch 14-16

In this talk I will present the derivation of the conserved energies which are equivalent to the H^s-norms of the solutions of the one-dimensional cubic nonlinear Schrödinger equations (NLS), that is, all the H^s-norms (with some lower bound for s) of the solutions are conserved a priori: This is done in the recent exciting work by Koch-Tataru. I will also present briefly my recent progress with Professor Koch for the Gross-Pitaevskii equation (GP): (GP) is defocusing (NLS) but with a non standard constraint on the solution q at infinity: |q| -> 1 at infinity. Technically this talk will be divided into three parts. In the introduction part, I will start with a rough statement about the conserved energies for (NLS) and (GP), and then introduce the essential notation: the transmission coefficient, and the three technical tools: the Hopf algebra, the frequency-rescaled norms and the superharmonic functions on the upper half plane. Then I will explain how to formulate the conserved energies by use of the transmission coefficient and how to show its equivalence to the H^s-norms by use of the technical tools: This is done by Koch-Tataru for (NLS). Finally for the Gross-Pitaevskii equation (GP) we will see that the non standard constraint at infinity causes essential new difficulties in the analysis.

11.07.2017
10:30 Uhr
Jean Bellissard
University of Münster, retired from Georgia Institute of Technology
Special Joint Analysis Seminar (Math+Physics):
A Toy Model for Viscosity
Hörsaal SeMath, Pontdriesch 14-16

A short review of the temperature behavior of liquids viscosity will be provided. The concept of anankeon as a new degree of freedom will be described and its relation with elastic degrees of freedom discussed. Then a simplistic solvable model, based on a Stochastic Markov dynamics, will be proposed and the solution explained and discussed. One consequence is the prediction that in a certain subclass of liquids, there is a bifurcation leading to a new time scale, the Maxwell time, which is liable to explain the exponential increase of the viscosity near the liquid-solid transition. A comparison with numerical simulations using molecular dynamics will be discussed.

16.05.2017
10:30 Uhr
Antione Hocquet
TU Berlin
Oberseminar:
Finite-time singularities of the stochastic harmonic map flow on surfaces.
Raum 001, Pontdriesch 14-16

A ferromagnetic material possesses a magnetization, which, out of equilibrium, satisfies the Landau-Lifshitz-Gilbert equation (LLG). Thermal fluctuations are taken into account by Gaussian space-time white noise. At least in the deterministic case, there is an important parallel between this model and the so-called Harmonic Map Flow (HMF). This was originally used by geometers (in the early sixties) as a tool to build harmonic maps between two manifolds u:M->N. The case where M is two dimensional is critical, in the sense that the natural energy barely fails to give well-posedness. We do not address here the problem of the solvability of LLG driven by space-time white noise. Instead, we consider a spatially correlated version. We show that contrary to the deterministic case, blow-up of solutions happen no matter how we choose the initial data.

23.05.2017
10:30 Uhr
Etienne Sandier
Université Paris 12
Joint Analysis Seminar:
Lower bound for energy growth of locally minimizing solutions of -Δu=u(1-|u|2) for u: ℝ3→ℝ2
Hörsaal SeMath, Pontdriesch 14-16

t.b.a.

13.06.2017
10:30 Uhr
Prof. Helmut Abels
Universität Regensburg
Joint Analysis Seminar:
Diffuse Interface Models for Two-Phase Flows of Incompressible Fluids and Their Sharp Interface Limits
Hörsaal SeMath, Pontdriesch 14-16

t.b.a.

09.05.2017
10:30 Uhr
Guido de Philippis
SISSA, Triest
Joint Analysis Seminar:
On the structure of measures satisfying a PDE constraint
Hörsaal SeMath, Pontdriesch 14-16

After a general introduction concerning the study of the interplay between PDE constraint and concentration/oscillation, I will present a general structure theorem for the singular part of Radon measure satisfying a PDE constraint. I will then present some applications.

02.05.2017
10:30 Uhr
David Gross
Universität zu Köln
Joint Analysis Seminar:
Low rank matrix recovery, the Clifford group, and some quantum mechnics
Hörsaal SeMath, Pontdriesch 14-16

t.b.a.

4-6 April, 2017
Aachen-Augsburg-Applied-Analysis Workshop
This two and a half day event explores recent progress in applied and stochastic analysis. Pontdriesch 14-16

Details at the webpage listed above.

07.02.2017
14:15 Uhr
Giuseppe Savaré
University of Pavia
Joint Analysis Seminar:
Singular perturbation of gradient flows and rate-independent evolution problems
klPhys, Schinkelstr. 2

t.b.a.

31.01.2017
14:15 Uhr
Manon Baudel
University of Orléans
Oberseminar:
Spectral theory for random Poincaré maps
Raum 001, Pontdriesch 14-16

We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting N asymptotically stable periodic orbits. To quantify the rare transitions between periodic orbits, we construct a discrete-time, continuous-space Markov chain, called a random Poincaré map. We show that this process admits exactly N eigenvalues which are exponentially close to 1, and provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighbourhoods of periodic orbits. The eigenvalues and eigenfunctions are well-approximated by principal eigenvalues and quasistationary distributions of processes killed upon hitting some of these neighbourhoods. The proofs rely on Feynman–Kac-type representation formulas for eigenfunctions, Doob’s h-transform, spectral theory of compact operators, and a detailed balance property satisfied by committor functions. Joint work with Nils Berglund (Orléans).

24.01.2017
14:15 Uhr
Christian Mießen
IMM RWTH Aachen
Oberseminar:
A massive parallel simulation approach to 2D and 3D grain growth in polycrystalline materials
Raum 001, Pontdriesch 14-16

We developed a highly efficient simulation model for 2D and 3D grain growth and recrystallization based on the level-set method. The developed model introduces modern computational concepts to achieve excellent performance on parallel computer architectures. We found strong scalability on ccNUMA architectures. For this purpose, the model considers the application of local level-set functions at the grain level. The model was utilized to simulate ideal and non-ideal grain growth in 2D and 3D with the objective to study the evolution of statistical representative volume elements in polycrystals. In addition, we simulated microstructure evolution in an anisotropic magnetic material affected by an external magnetic field.

20.12.2016
14:15 Uhr
Sebastian Scholtes
RWTH Aachen University
Oberseminar:
Variations on geometric knot theory
Raum 001, Pontdriesch 14-16

The talk will encompass several themes in geometric knot theory. We will talk about new geometric curvature energies and the discretisation of some established energies. For the last item, we consider different modes of variational convergence. Time allowing, we explore the connection of geometric curvature energies to discrete and metric geometry.

13.12.2016
14:15 Uhr
Giovanni Di Fratta
CMAP, Ecole Polytechnique
Oberseminar:
Thin convex shells in micromagnetics
Raum 001, Pontdriesch 14-16

The talk is devoted to the Γ-development analysis of the micromagnetic energy functional, when the domain occupied by the nanomagnet is a thin shell generated by a bounded and convex smooth surface. Indeed, recently a significant interest to nanomagnets with curved shape has appeared. In particular, spherical shells are currently of great interest due to their capability to support skyrmion solutions which can be stabilized by curvature effects only, in contrast to the planar case where the intrinsic Dzyaloshinsky-Moriya interaction is required. It is well established that the effects of the demagnetizing field operator can be reduced to an effective easy-surface anisotropy for planar thin shells whose thickness is much less than the size of the system. More precisely, in G. Gioia and R. D. James (Micromagnetics of very thin films. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 453(1956):213–223, 1997) it is shown that the effects of the demagnetizing field operator can be reduced to an effective easy-surface anisotropy for planar thin shells. A generalization of this result can be found in G. Carbou (Thin layers in micromagnetism. Mathematical Models and Methods in Applied Sciences, 11(09):1529–1546, 2001) where, for thin shells generated by extruding surfaces whose closure is diffeomorphic to the closed unit disk of R^2, the asymptotic behavior of the minimizers of the micromagnetic energy functional is investigated. In V. Slastikov (Micromagnetics of thin shells. Mathematical Models and Methods in Applied Sciences, 15(10):1469–1487, 2005) a Γ-convergence analysis is performed on pillow-like thin shells. In all the cited cases the investigation leaves out very interesting scenarios like the spherical one which cannot be easily recovered by simply gluing local patches. In this talk we present, in the spirit of the theoretical framework presented in G. Anzellotti and S. Baldo (Asymptotic development by Γ-convergence. Applied mathematics and optimization, 27(2):105–123, 1993), a Γ-development analysis of the micromagnetic energy functional when the shell is generated, like in the case of a sphere, by a bounded and convex smooth surface.

06.12.2016
14:15 Uhr
John M. Ball
University of Oxford
Joint Analysis Seminar:
The mathematics of liquid crystals
klPhys, Schinkelstr. 2

t.b.a.

02.12.2016
14:15 Uhr
Sören Bartels
Universität Freiburg
Mathematisches Kolloquium:
Approximation of large bending problems
Hörsaal SeMath, Pontdriesch 14-16

The development of polymer structures suggests various new applications in the area of nanotechnology. The controlled fabrication of related devices and nanotools leads however to many difficulties. Numerical simulations can contribute to improving this. In the talk we discuss the mathematical modeling and reliable computation of large bilayer bending effects. Deformations are described via a nonlinear bending energy subject to a pointwise isometry constraint. We devise finite element discretizations using discrete Kirchhoff triangles and show accuracy of approximations via Gamma-convergence of the discretized functionals. The practical energy minimization is based on a semiimplicit discretization of a related gradient flow. Self-avoidance of deformations is not included in the model and appears to be relevant only in some situations. We present first results concerning the convergent computation of self-avoiding inextensible curves using a tangent-point functional.

18.11.2016
14:15 Uhr
Stefan Müller
Universität Bonn
Mathematisches Kolloquium:
The mathematics of packing, crumpling and folding
Hörsaal SeMath, Pontdriesch 14-16

A number of questions in physics and biology involve the problem of packing thin elastic objects in a container in an optimal way, e.g. with minimal elastic energy. In this lecture I will discuss precise mathematical formulations of some of these problems, recent results and open conjectures

17.11.2016
11:00 Uhr
Zisis N. Sakellaris
Athen
Oberseminar:
Minimization of Curvature in Conformal Geometry
Raum 001, Pontdriesch 14-16

t.b.a.

15.11.2016
14:15 Uhr
Qiji J. Zhu
Western Michigan University
Joint Analysis Seminar:
Variational methods in the presence of symmetry
klPhys, Schinkelstr. 2

t.b.a.

25.10.2016
14:15 Uhr
Gero Friesecke
TU München
Joint Analysis Seminar:
Inferring atomic structure from X-ray diffraction patterns
klPhys, Schinkelstr. 2

t.b.a.

05.07.2016
10:45 Uhr
Andrea Mondino
ETH Zürich
Joint Analysis Seminar:
Non-smooth spaces with Ricci curvature lower bounds
Hörsaal SeMath, Pontdriesch 14-16

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80s and was pushed by Cheeger and Colding in the '90s who investigated the structure of the spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can a give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, etc.), permitting to give further insights in the theory. In the seminar I will give an overview of the topic. In the talk, after a brief introduction to the topic I will present some recent results.

14.06.2016
10:45 Uhr
Camilla Nobili
University of Basel
Oberseminar:
Bounds on turbulent convection
Raum 001, Pontdriesch 14-16

We present some recent results regarding the Rayleigh-Bénard convection. In a first part we deal with the infinite Prandtl number limit of the Boussinesq equations and we introduce the temperature background field method in order to find upper bounds on the average upward heat transport. Inspired by the seminal result of Doering, Otto and Westdickenberg (née Reznikoff) in 2007 we characterized all admissible background profiles and we show that the upper bound of Otto and Seis 2013 is optimal within this method. In the second part we consider fluids with finite Prandtl number and we present a new upper bound on the Nusselt number which catches a transition in the phase space (Pr, Ra).

07.06.2016
10:45 Uhr
Robert L. Jerrard
University of Toronto
Joint Analysis Seminar:
Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii equation
Hörsaal SeMath, Pontdriesch 14-16

TBA

24.05.2016
10:45 Uhr
Eric A. Carlen
Rutgers University
Joint Analysis Seminar:
The structure of near minimizers of a non-local free energy functional
Hörsaal SeMath, Pontdriesch 14-16

The Gates-Pentrose-Lebowitz free energy functional is a non-local analog of the phenomenological van der Waals free energy function. Unlike the local van der Waals functional, it arises in the continuum limit of lattice gas models. As in many statistical mechanical models in which free energy functionals are large deviations functional, one is interested not only in the absolute minimizers, but also the near minimizers. In classical work on the near minimizers of the van der Waals functional, the co-area formula and the quantitative isoperimetric inequality play a crucial role. We present joint work with Maggi on a non-local quantitative isoperimetric inequality, and discuss its application to the description of near minimizers of the Gates-Pentrose-Lebowitz free energy functional in work with Carvalho, Esposito, Marra and Lebowitz.

03.05.2016
10:45 Uhr
Yuriy Mokrousov
FZ Jülich
Oberseminar:
Topological Spintronics: guided by ab-initio research
Raum 001, Pontdriesch 14-16

Currently, solid state physics experiences a revolution associated with the advent of novel topological concepts. It was realized that, as by magic, many of the well-known phenomena can suddenly be intuitively understood following abstract mathematical ideas which tell us that the dynamics of electrons in solids is directly related to topology of the spaces in which they live. This revolution has completely changed the landscape of spintronics, which deals with the interplay between spin and charge of electrons, leading to predictions of novel, topological states of matter such as topological insulators and magnetic skyrmions. In my talk, on simple but important examples, I will explain how the topological standpoint can be utilized for the prediction of novel effects related to electron dynamics, and lead to quantitative description of these effects in most complex materials.

26.04.2016
10:45 Uhr
Dorin Bucur
Université de Savoie
Joint Analysis Seminar:
Optimal shapes and isoperimetric inequalities for spectral functionals
Hörsaal SeMath, Pontdriesch 14-16

In this talk I will discuss isoperimetric inequalities involving the spectrum of the Laplace operator (of Faber-Krahn or Saint-Venant type) seen from the perspective of shape optimization. I will focus on problems involving the spectrum of the Robin-Laplacian and the Steklov problem, and show techniques developed around the Mumford-Shah functional in image segmentation theory can be used to prove such inequalities.

19.04.2016
14:00 Uhr
Carina Geldhauser
Universität Bonn
Special Seminar:
The $L^2$ gradient flow of the double well potential and a related interacting particle system
Raum 001, Pontdriesch 14-16

We consider the forward-backward parabolic equation $u_t = \left( W'(u_x)\right)_X$ with periodic boundary conditions in one space dimension, where $W$ is a double well potential. This equation is the formal $L^2$ gradient flow of the non-convex potential $W(p) = \frac{1}{4} (1-p^2)^2$, which is a priori ill-posed. The first part of the talk will show how regularisation by discretisation in space can be used to prove existence of solutions of the equation for a special class of initial data. In the second part of the talk, we view such discrete-in-space schemes as systems of particles driven by the potential $W$ and add a perturbation by independent Brownian motions to their dynamics. We address the question how many particles are allowed to interact with each other such that the resulting interacting particle system converges as $h\to 0$ to a well-posed stochastic PDE. If time allows, we will state some results on the long-time behaviour of solutions to the limit SPDE. This talk combines results obtained in collaboration with Giovanni Bellettini, Anton Bovier and Matteo Novaga.

19.04.2016
10:45 Uhr
Melanie Rupflin
University of Oxford
Oberseminar:
Flowing to minimal surfaces
Raum 001, Pontdriesch 14-16

For maps from surfaces there is a close connection between the area functional and Dirichlet energy and thus also between their critical points. As such, one way to try to find critical points of the Area is to consider a gradient flow of the Dirichlet energy, which not only evolves a map but also the domain metric in order to find a map that is not only harmonic but also (weakly) conformal and thus a (branched) minimal immersion. In this talk I will discuss the construction of such a flow, the Teichmueller harmonic map flow, and explain in particular how this flow decomposes any given initial map into one or more branched minimal immersions.

09.02.2016
14:15 Uhr
MPI MIS Leipzig
Oberseminar:
An epiperimetric inequality for the thin obstacle problem
Raum 001, Pontdriesch 14-16

I will discuss some recent results on the thin obstacle problem (also known as the scalar Signori problem). This is a classical problem in the calculus of variations and, in its simplest form, it consists in minimizing the Dirichlet energy among all functions that are constrained to have a positive trace on a hyperplane.

Despite the long tradition, many questions concerning the regularity of the solutions remain still open. In this talk, our focus is on the rate of converge of the solutions to their unique blowup limits near certain points of the free boundary. Following the pioneering work by G. Weiss on the classical obstacle problem, we prove an "epiperimetric inequality" a la Reifenberg, which closely resembles the one firstly shown in the '60s in the study of the regularity of minimal surfaces.

This is a joint work with M. Focardi of the University of Florence.

26.01.2016
14:15 Uhr
Alexander Lytchak
University of Cologne
Joint Analysis Seminar:
Classical Plateau Problem in metric spaces
klPhys

TBA

19.01.2016
14:15 Uhr
Florian Nolte
Universität Heidelberg
Oberseminar:
Domains in ferromagnetic films with perpendicular anisotropy
Raum 001, Pontdriesch 14-16

We consider a regime of ultra-thin ferromagnetic films with strong anisotropy and easy axis pointing out of the film plane. Experiments show the formation of various domain patterns such as so-called bubble or maze domains. We aim to explain some of these patterns using an energy minimization approach - starting from the full three-dimensional micromagnetic model, we identify a reduced regime and derive the asymptotic behavior of the energy in the framework of $\Gamma$-convergence. Our analysis indicates that the typical length scale of domains grows exponentially as a function of the inverse thickness of the plate.
When the ferromagnetic film is exposed to a critical external field, Knüpfer and Muratov have shown that a branching domain pattern consisting of long and slender domains is (in terms of scaling) energetically optimal. We study a single of those long and slender domains using a toy model: the perimeter functional with an additional nonlocal term modelling dipole interaction. We show existence, regularity and connectedness of minimizers as well as the scaling of the minimal energy in terms of the prescribed volume of the domain.

15.12.2015
14:15 Uhr
Laure Saint-Raymond
Pierre et Marie Curie University (Paris VI)
Joint Analysis Seminar:
From systems of particles to kinetic and fluid models
klPhys

TBA

11.12.2015
14:15 Uhr
Tim Laux
MPI MIS Leipzig
Oberseminar:
Convergence of thresholding schemes for geometric flows
Raum 001, Pontdriesch 14-16

The thresholding scheme, a time discretization for mean-curvature flow was introduced by Meriman, Bence and Osher in 1992. In the talk we present new convergence results for modern variants of this scheme, in particular in the multi-phase case with arbitrary surface tensions. The first result establishes convergence towards a weak formulation of mean-curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movement scheme by Esedoglu and Otto in 2014. This interpretation means that the thresholding scheme preserves the structure of (multi-phase) mean-curvature flow as a gradient flow w. r. t. the total interfacial energy. More precisely, the thresholding scheme is a minimizing movement scheme for an energy that $\Gamma$-converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren, Taylor and Wang in 1993 and Luckhaus and Sturzenhecker in 1995, which establish convergence of a more academic minimizing movement scheme. Like the one of Luckhaus and Sturzenhecker, ours is a conditional convergence result, which means that we have to assume that the time-integrated energy of the approximation converges to the time-integrated energy of the limit. This is a natural assumption, which is however not ensured by the compactness coming from the basic estimates. We will also discuss new convergence results for volume-preserving mean-curvature flow and forced mean-curvature flow.
--- Based on joint works with Felix Otto (MPI MIS Leipzig) and Drew Swartz (Purdue University)

24.11.2015
14:15 Uhr
Tim Sullivan
University of Warwick
Joint Analysis Seminar:
Brittleness and robustness of Bayesian inference
klPhys

The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the frequentist questions of well-specification and consistency, or the numerical analysis questions of stability and well-posedness with respect to perturbations of the prior, the likelihood, or the data. This talk will outline positive and negative results (both classical ones from the literature and new ones due to the authors and others) on the accuracy of Bayesian inference. There will be a particular emphasis on the consequences for high- and infinite-dimensional complex systems. In particular, for such systems, subtle details of geometry and topology play a critical role in determining the accuracy or instability of Bayesian procedures. Joint work with Houman Owhadi and Clint Scovel (Caltech).

10.11.2015
14:15 Uhr
Eduard Feireisl
Academy of Sciences of the Czech Republic
Joint Analysis Seminar:
On solvability of certain problems in fluid mechanics involving inviscid fluids
klPhys

We present a general method of "construction" of solutions to systems of partial differential equations describing the motion of inviscid fluids. The method is based on a variable coefficients version of the oscillatory lemma proved in the context of the incompressible Euler system by C. De Lellis and L. Székelyhidi. Several specific examples of fluid systems including the Korteweg fluid models, quantum fluids, the Euler-Fourier system, and the Savage-Hutter model will be presented. We also discuss suitable admissibility criteria to ensure well-posedness of these problems.

03.11.2015
14:15 Uhr
Benjamin Berkels
RWTH Aachen
Oberseminar:
Variational approaches for image segmentation and registration
Raum 001, Pontdriesch 14-16

Image segmentation and registration are two of the fundamental image processing problems. Segmentation is to decompose an image into disjoint regions that are roughly homogeneous in a suitable sense. If three or more regions are sought, one speaks of multi-phase segmentation. We first review how to find global minimizers of the non-convex binary Mumford-Shah model to solve the classical two-phase segmentation problem. Then, we propose a novel method for multi-phase segmentation of images based on the Mumford-Shah model and high-dimensional local feature vectors. While the method was developed for the segmentation of extremely noisy crystal images based on localized Fourier transforms, the resulting framework is not tied to specific feature descriptors. For instance, using local spectral histograms as features, it allows for robust texture segmentation.

Image registration is the task of transforming two or more images into a common coordinate system. After a short introduction to variational image registration, we discuss how a series of hundreds of noisy scanning transmission electron microscopy (STEM) images can be registered to obtain an improved image that enables sub-picometer precision measurements of atom positions surpassing the quality attainable by single shot STEM images. Particular challenges here are input data with a low signal-to-noise ratio and periodic structures. Finally, non-rigid approaches in the context of medical imaging are discussed.

13.10.2015
15:00 Uhr
Stavros Komineas
University of Crete
Oberseminar:
Dynamics of skyrmions in chiral ferromagnets
Raum 001, Pontdriesch 14-16

We will give a description of the dynamics of topological and non-topological solitons in ferromagnetic films. We study materials with a Dzyaloshinskii-Moriya interaction and easy-axis anisotropy. Our analysis is based on an important link between topology and dynamics which is established through the construction of unambiguous conservation laws. In particular, we study the motion of a topological skyrmion with skyrmion number Q=1 and a non-topological skyrmionium with Q=0 under the influence of an applied field gradient (which plays the role of a force). The Q=1 skyrmion undergoes Hall motion perpendicular to the direction of the field gradient with a drift velocity proportional to the gradient. In contrast, the non-topological Q=0 skyrmionium is accelerated in the direction of the field gradient, thus exhibiting ordinary Newtonian motion. When the applied field is switched off the Q=1 skyrmion is spontaneously pinned around a fixed guiding center, whereas the Q=0 skyrmionium moves with constant velocity v. We give a numerical calculation of a skyrmionium traveling with any constant velocity v that is smaller than a critical velocity vc.

14.07.2015
14:00 Uhr
John McCuan
Georgia Tech
Oberseminar:
Numerical limits of stability for cylindrical pendent drops
Raum 001, Pontdriesch 14-16

Cylindrical pendent drops are a hybrid between pendent drops and horizontal liquid bridges in a downward gravity field.  They are somewhat remarkable for admitting discontinuous stable families indexed by increasing volume and stable drops with volume larger than some unstable drops adhering to the same support geometry in particular.  This feature, among others, complicates the stability analysis, and while some stability properties of these drops can be rigorously proved, some assertions presently depend on numerical calculations.  In this talk I will describe the basic stability behavior distinguishing between rigorous and numerical results and giving particular emphasis to some recently discovered fine properties exposed by the numerics.  Time permitting, I will discuss details of our numerical approach and some technical aspects relying on asymptotic analysis.

23.06.2015
14:00 Uhr
Patrick Dondl
Durham University
Oberseminar:
Motion of interfaces in random media: pinning and some applications
Raum 001, Pontdriesch 14-16

We consider the evolution of an interface, subject to a driven mean curvature flow, in a random environment. The environment is modeled by a non-linear, random, forcing term in the evolution equation and describes localized obstacles which are harder to penetrate by the interface. First we will consider a the problem of pinning a nearly flat interface in such a random field of obstacles, proving existence of a stationary solution of the evolution equation by a combination of percolation results and sub- and supersolution techniques. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force). We will then apply some of these ideas to solutions of Landau-de Gennes' theory of nematic liquid crystals in the sharp interface limit, considering the evolution of interfaces with spherical initial conditions.

02.06.2015
14:00 Uhr
Benjamin Schlein
University of Zurich
Joint Analysis Seminar:
Nonsmooth differential geometry. Abstract at Instmath event page.
Hörsaal SeMath, Pontdriesch 14-16

Abstract at Instmath event page.

12.05.2015
14:00 Uhr
Karlheinz Gröchenig
University of Vienna
Joint Analysis Seminar:
Deformation of Gabor systems. Abstract at Instmath event page.
Hörsaal SeMath, Pontdriesch 14-16

Abstract at Instmath event page.

05.05.2015
14:00 Uhr
Aldo Pratelli
University of Erlangen-Nürnberg
Joint Analysis Seminar:
On the approximation of Sobolev homeomorphisms. Abstract at Instmath event page.
Hörsaal SeMath, Pontdriesch 14-16

Abstract at Instmath event page.

21.04.2015
14:00 Uhr
Nicola Gigli
University Pierre et Marie Curie (Paris VI)
Joint Analysis Seminar:
Nonsmooth differential geometry. Abstract at Instmath event page.
Hörsaal SeMath, Pontdriesch 14-16

Abstract at Instmath event page.

03.03.2015
14:00 Uhr
Yuko Nagase
Osaka City University Advanced Mathematical Institute (OCAMI)
Oberseminar:
Analysis of a Cahn-Hilliard/Allen-Cahn equation
Raum 001, Pontdriesch 14-16

In this talk I will present some analyses of a Cahn-Hilliard/Allen-Cahn equation which was introduced by Karali and Katsoulakis. The limit evolution of the equation under a suitable scaling is also mean curvature flow, similarly to the Allen-Cahn equation but a different coefficient constant appears, which speeds up the mean curvature flow. We will show some analyses, that is, existence of solution and asymptotic behaviors. As next step, we will consider a stochastic version of the equation. I will present some recent known results of the existence of solution and related results.

10.02.2015
15:00 Uhr
Peter Hornung
TU Dresden
Oberseminar:
The Willmore functional on isometric immersions
Raum 001, Pontdriesch 14-16

In this talk we present some results about the (generalised) Kirchhoff plate functional from nonlinear elasticity. This functional is obtained by restricting the classical Willmore functional to the class of isometric immersions of a fixed 'reference' Riemannian manifold.

We will present a general framework to study such functionals, and we will construct a 'reference' metric whose associated Kirchhoff functional admits infinitely many stationary points.

20.01.2015
16:00 Uhr
Laurent Demanet
MIT
Joint Analysis Seminar:
1930s analysis for 2010s signal processing: recent progress on the super-resolution question. Abstract at Lehrstuhl C event page.
Hörsaal SeMath, Pontdriesch 14-16

Abstract at Lehrstuhl C event page.

13.01.2015
16:00 Uhr
Alexei N. Bogdanov
IFW Dresden
Special seminar:
Chiral skyrmions: rigorous solutions and physical applications
Hörsaal SeMath, Pontdriesch 14-16

Two dimensional chiral magnetic vortices or skyrmions represent areas of reverse magnetization localized into tubes with the diameters of nanoscale sizes. Importantly that in most of nonlinear physical systems such multidimensional static solitonic states are unstable and collapse spontaneously under the influence of external or internal perturbations. In condensed matter systems lacking inversion symmetry two- and three-dimensional localized states (skyrmions) are stabilized by a specific mechanism imposed by handedness of the underlying structure [1]. This single out condensed matter systems with intrinsic and induced chirality (noncentrosymetric magnetic crystals, multiferroics, ferroelectrics, and liquid crystals) into a particular class of materials where skyrmions can be induced and manipulated.

I overview theoretical results for isolated and embedded skyrmions arising in bulk and confined magnetic materials with intrinsic and induced chirality [1-4].

[1] A. Bogdanov, A. Hubert. JMMM 138, 255 (1994); 195, 182 (1999).
[2] M. N. Wilson et al., PRB 89, 094411 (2014).
[3] F. N. Rybakov et al. Phys.Rev. B 87, 094424 (2013).
[4] S. A. Meynell et al., Phys.Rev. B 90, 014406 (2014).

06.01.2015
16:00 Uhr
Felix Otto
MPI for Mathematics in the Sciences
Joint Analysis Seminar:
A regularity theory for elliptic equations with random coefficients. Abstract at Lehrstuhl C event page.
Hörsaal SeMath, Pontdriesch 14-16

Abstract at Lehrstuhl C event page.

18.11.2014
16:00 Uhr
Martin Rumpf
University of Bonn
Joint Analysis Seminar:
Time Discrete Geodesics on Image Manifolds. Abstract at Lehrstuhl C event page.
Hörsaal SeMath, Pontdriesch 14-16

The abstract is posted at http://www.mathc.rwth-aachen.de/news/home/.

11.11.2014
15:00 Uhr
University of Bremen
Oberseminar:
Pattern formation in simple spintronic device models with aligned fields.
Raum 001, Pontdriesch 14-16

The self-organized emergence of spatio-temporal patterns is a ubiquitous phenomenon in nonlinear processes on large homogeneous domains. In this talk a class of Landau-Lifshitz-Gilbert-Slonczewski equations is studied from this viewpoint, highlighting various aspects of the theory. The model describes magnetization dynamics in the presence of an applied field and a spin polarized current. Here we consider the case of axial symmetry and focus on coherent structure solutions that occur due to the symmetry. This is joint work with Christof Melcher (RWTH).

04.11.2014
18:00 Uhr
Angela Stevens
University of Münster
Colloquium:
Mathematical Modeling of Structure Formation in Cellular Systems due to Cell Motion
Hörsaal III

Chemotaxis - directed cell motion due to attractive, mostly diffusible, signals - is a common mechanism for pattern and structure formation in developmental cell systems. Related mathematical models are partial differential equations of cross-diffusion type. Signals for cell communication can also be more localized (e.g. within the so-called extra cellular matrix). In this case mathematical models of partial differential equations coupled with ordinary differential equations result. Pattern formation of both kinds of models differ and are discussed. Also mathematically different techniques are required to analyze these models w.r.t. their pattern forming behavior. Further, there are connections with self-attracting reinforced random walks, which will also be discussed.

28.10.2014
16:00 Uhr
Camillo DeLellis
University of Zurich
Joint Analysis Seminar:
Dissipative continuous solutions of the incompressible Euler equations with Onsager-critical spatial regularity. Abstract at Lehrstuhl C event page.
Hörsaal SeMath, Pontdriesch 14-16

The abstract is posted at http://www.mathc.rwth-aachen.de/news/home/.

13.10.2014
16:00 Uhr
Martin Hairer
Warwick University
Joint Analysis Seminar:
Weak universality of the KPZ equation. (Note this talk is on a Monday!) Abstract at Lehrstuhl C event page.
Hörsaal SeMath, Pontdriesch 14-16

The abstract is posted at http://www.mathc.rwth-aachen.de/news/home/.

15.07.2014
16:00 Uhr
Zdzislaw Brzezniak
University of York
Joint Analysis Seminar:
See Lehrstuhl C event page
I

The abstract is posted at http://www.mathc.rwth-aachen.de/news/home/.

08.07.2014
15:00 Uhr
Lucia Scardia
University of Glasgow
Oberseminar: POSTPONED SG 11

The seminar is postponed. A new date will be announced next semester.

01.07.2014
16:00 Uhr
Felix Schulze
University College London
Joint Analysis Seminar:
See Lehrstuhl C event page
I

The abstract is posted at http://www.mathc.rwth-aachen.de/news/home/.

17.06.2014
16:00 Uhr
Massimo Fornasier
TU Muenchen
Joint Analysis Seminar:
See Lehrstuhl C event page
I

The abstract is posted at http://www.mathc.rwth-aachen.de/news/home/.

13.05.2014
16:00 Uhr
Changyou Wang
University of Kentucky
Joint Analysis Seminar:
CANCELLED
I
06.05.2014
16:00 Uhr
Mauro Maggioni
Duke University
Joint Analysis Seminar:
See Lehrstuhl C event page
I

The abstract is posted at http://www.mathc.rwth-aachen.de/news/home/.

29.04.2014
15:00 Uhr
Paola Pozzi
University of Duisburg-Essen
Oberseminar: On the elastic flow for open curves SG 11

In this talk I will discuss a long-time existence result for the elastic flow of open curves in $\mathbb{R}^n$.

15.04.2014
15:00 Uhr
Lukas Döring
MPI Leipzig
Oberseminar: Reduced models for domain walls in soft ferromagnetic films SG 11

Depending on material parameters and sample dimensions, soft ferromag- netic films exhibit various kinds of domain walls (transition layers between neighboring domains of constant magnetization). In the first part of the talk, we present a reduced model for a single domain wall at the cross-over from symmetric to asymmetric wall types in an infinitely extended film. The reduced model captures the optimal splitting of the wall into an asymmetric, stray-field free core and logarithmically decaying, symmetric wall tails (obtained via Gamma conver- gence, joint work with R. Ignat and F. Otto). In the second part, we consider periodic domain patterns for which the wall width is not only confined by material anisotropy but also the neighboring wall tails. Using methods similar to the single-wall case, we derive a reduced model that describes the influence of the tail interaction on the splitting into the core and tails. Moreover, it yields a prediction for the increased rotation of the magnetization towards magnetic fields applied along the wall normal direction.

04.02.2014
15:00 Uhr (Note there is a second talk this day)
Barbara Niethammer
University of Bonn
Joint Analysis Seminar: Phase transitions in a nonlocal Fokker-Planck equation with multiple time scales Fo 3

04.02.2014
16:45 Uhr (Note special time)
Alexander Mielke
Weierstrass Institute for Applied Analysis and Stochastics (WIAS)
Joint Analysis Seminar: On gradient structures for reaction-diffusion systems Fo 3

28.01.2014
15:00 Uhr
Peter Sternberg
Indiana University
Oberseminar: Minimizers of a nonlocal isoperimetric problem in thin domains Fo 3

The nonlocal isoperimetric problem arises as a limit of the Ohta-Kawasaki model for diblock copolymers. As a variational problem, it takes the form of a competition between a local term favoring low surface area and a nonlocal term favoring high oscillation. In this talk I will survey some of the previous activity on this problem (of which there is a lot) and then focus on a recent result by Massimiliano Morini and me in the setting of thin domains where we can identify the global minimizer for all values of the coefficient of nonlocality.

07.01.2014
15:00 Uhr
Thierry Gallay
University of Grenoble
Joint Analysis Seminar: Distribution of energy and convergence to equilibria in extended dissipative systems Fo 3

We study the local energy dissipation in gradient-like nonlinear partial differential equations on unbounded domains. Our basic assumption, which happens to be satisfied in many classical examples, is a pointwise upper bound on the energy flux in terms of the energy dissipation rate. Under this hypothesis, we derive a simple and general bound on the integrated energy flux which implies that, in low space dimensions, our extended dissipative system'' has a gradient-like dynamics in a suitable averaged sense. In particular, we can estimate the time spent by any trajectory outside a neighborhood of the set of equilibria. As an application, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equation in an infinite cylinder. This talk is based on a collaboration with S. Slijepcevic (Zagreb, Croatia).

10.12.2013
15:00 Uhr
Didier Bresch
University of Savoie
Joint Analysis Seminar: Mathematical topics around shallow-water type equations Fo 3

29.11- 30.11.2013 Analysis Day
Speaker and schedule information for this two day event

19.11.2013
15:30 Uhr
Martina Hofmanova
MPI Leipzig
Oberseminar: The concept of kinetic solutions and well-posedness for stochastic conservation laws SG 11

Many basic equations in physiscs can be written in the form of conservation law. However, as it is common in the field of PDEs and SPDEs, classical or strong solutions do not exist in general and, on the other hand, weak solutions are not unique. The notion of kinetic formulation and kinetic solution turns out to be a very convenient tool to overcome these difficulties.
In this talk, I will show how this approach can be adapted to stochastic problems and present several well-posedness results in the case of hyperbolic, semilinear degenerate parabolic and quasilinear degenerate parabolic setting. The talk is based on joint works with Arnaud Debussche and Julien Vovelle.

05.11.2013
15:30 Uhr
Peter Bella
MPI Leipzig
Oberseminar: Wrinkling of a Stretched Annular Elastic Thin Sheet - Identification of the Optimal Scaling Law for the Ground State Energy SG 11

In [Bella & Kohn: Wrinkles as the result of compressive stresses in an annular thin film, to appear in CPAM] we identified the optimal scaling law of the minimum of the elastic energy of a stretched annular thin elastic sheet. In this talk I will describe the next step towards the understanding of this problem -- I will identify the optimal prefactor in the scaling law. Moreover, I show that this prefactor can be characterized as a minimum of a much simpler (scalar) variational problem.

17.12.2013
15:30 Uhr
Filip Rindler
University of Warwick
Oberseminar: Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms SG 11

Microlocal compactness forms (MCFs) are a new tool to study oscillations and concentrations in L^p-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending both the theory of (generalized) Young measures and the theory of H-measures. Since in L^p-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, MCFs allow to quantify the difference between these two notions of compactness. The definition involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar's framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a "geometric" way. Moreover, the hierarchy of oscillations with regard to slow and fast scales can be investigated as well since this information is also is reflected in the generated MCF.

22.10.2013
15:00 Uhr
Massimo Fornasier
Technical University München
CANCELLED!: Joint Analysis Seminar on Quasi-linear compressed sensing Fo 3

Unfortunately this talk was cancelled due to illness. Please feel free to join anyway for coffee at 14:15 in room 114 in the Main Building.

16.07.2013
18:00 Uhr
Stanley Alama
McMaster University
Kolloquium: Vortices and the Ginzburg-Landau Functional Hörsaal III

The Ginzburg-Landau model is a widely used tool for describing the physical state of superconductors, superfluids, or Bose-Einstein condensates. It also appears in particle physics as the Abelian Higgs model. Ginzburg-Landau is a rich source of interesting phenomena in the calculus of variations and in the study of singularities in solutions to partial differential equations. In this talk, I will start with the basics of Ginzburg-Landau vortices, beginning with the classical results of Bethuel-Brezis-Hélein. Then I will present some theorems on vortex lines for the 3D Ginzburg-Landau model, in the context of Gamma-convergence. Finally, I will present an overview of some recent results on the effect of anisotropy in the mathematical study of superconductors. Anisotropic models are important for the understanding of the high-temperature superconductors, and they present some interesting mathematical results and challenges.

16.07.2013
15:45 Uhr
Lia Bronsard
McMaster University
Joint Analysis Seminar: Vortices for a 2 component Ginzburg Landau system Hörsaal V

We study vortices in a Ginzburg-Landau model for a pair of complex-valued order parameters.  Multi-component functionals have been introduced in the context of unconventional p-wave superconductors and spinor Bose-Einstein condensates to include spin coupling effects. As in the classical Ginzburg-Landau model, minimizers will exhibit quantized vortices in response to boundary conditions or applied fields.  However, we show that the interaction between the two components allows for vortices with a more exotic core structure.  Our results are based on a combination of variational and PDE methods, blowing up around the vortex core and studying the resulting system and its local minimizers. This is joint work with Stan Alama and Petru Mironescu.

09.07.2013
15:30 Uhr
Max Jensen
University of Sussex
Seminar: A Finite Element Method for Hamilton-Jacobi-Bellman equations SG 11

Hamilton-Jacobi-Bellman equations describe how the cost of an optimal control problem changes as problem parameters vary. This talk will address how Galerkin methods can be adapted to solve these equations efficiently. In particular, it is discussed how the convergence argument by Barles and Souganidis for finite difference schemes can be extended to Galerkin finite element methods to ensure convergence to viscosity solutions. A key question in this regard is the formulation of the consistency condition. Due to the Galerkin approach, coercivity properties of the HJB operator may also be satisfied by the numerical scheme. In this case one achieves besides uniform also strong H^1 convergence of numerical solutions on unstructured meshes.

11.06.2013
15:30 Uhr
Margaret Beck
Heriot-Watt University
Seminar: Quasi-stationary states of the 2D Navier-Stokes equation SG 11

The dynamics of the 2D incompressible Navier-Stokes equation has two key stages: rapid convergence to certain quasi-stationary states, and then the slow evolution of these states as the fluid relaxes to its rest state. We provide a mathematical explanation for these multiple time scales by studying a certain family of quasi-stationary states, known as bars and dipoles, using techniques from the theory of infinite-dimensional dynamical systems and hypocoercive operators.

18.06.2013
15:45 Uhr
Michael Struwe
ETH
Joint Analysis Seminar: The gradient flow for a supercritical elliptic variational problem Hörsaal V

04.06.2013
15:45 Uhr
Sergio Conti
Universität Bonn
Joint Analysis Seminar: Derivation of a line-tension model for dislocations in the plane Hörsaal V

Dislocations are topological defects in crystals which generate long-range elastic stresses. For dislocations in the plane the elastic interactions can be represented via a singular kernel behaving as the H^{1/2} norm of the slip. We obtain a sharp-interface limit within the framework of Gamma convergence in the limit of small elastic spacing. One key ingredient is a proof of the fact that the presence of infinitely many equivalent length scales gives strong restrictions on the geometry of the microstructure. In particular we show that the micrustructure must be one-dimensional on most length scales, and that only few are available for the relaxation. This talk is based on joint work with Adriana Garroni, Annalisa Massaccesi and Stefan Müller.

28.05.2013
15:45 Uhr
Mark Peletier
University of Eindhoven
Joint Analysis Seminar: Energy-driven pattern formation via competing long- and short-range interactions Hörsaal V

I will discuss patterns in block copolymer melts. This is a model system that is mathematically tractable, physically meaningful (and experimentally accessible) and representative for a large class of energy-driven pattern-forming systems. Such systems show a remarkable variety of different patterns, of which only a small fraction is well understood.
In this talk I will focus on a variational model for this system, in a parameter regime in which the system forms regular patterns of small spheroid blobs, called particles. The energy for these structures is dominated by a single-particle term, which penalizes each particle independently. This term drives the system towards particles of a well-defined size. At the next level the interaction between the particles is given by a Coulomb interaction potential, giving rise to approximately periodic arrangements.

07.05.2013
15:45 Uhr
Giuseppe Buttazzo
University of Pisa
Joint Analysis Seminar: Optimal potentials for Schrödinger operators Hörsaal V

We consider the Schrödinger operator $-\Delta+V(x)$ on $H^1_0(\mathcal{O})$, where $\mathcal{O}$ is a given domain of $\mathbb{R}^d$. Our goal is to study some optimization problems where an optimal potential $V\ge0$ has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

30.04.2013
15:30 Uhr
Angkana Rüland
Universität Bonn
Seminar: Rigidity, non-rigidity and scaling of the cubic-to-orthorhombic phase transition in the linear theory of elasticity SG 11

In this talk I present recent results on rigidity properties of the cubic-to-orthorhombic phase transition in the linear theory of elasticity. Using the framework of convex integration, it is proved that this model provides an example of a martensitic phase transition in which already in the linear theory of elasticity no rigidity properties can be expected without requiring additional regularity conditions on the phase interfaces.
As a complementary result, it is demonstrated that in the generic piecewise polygonal situation, i.e. if the phases are separated by a finite number of piecewise polygonal interfaces, the material is rigid. Hence, in this situation locally the only possible configurations are laminates and crossing twins. Finally, in a reduced model for the cubic-to-orthorhombic transition involving interfacial energy, a scaling result for crossing twin structures is derived. This reflects the stability of crossing twin structures under small perturbations. These results are part of the author's diploma and PhD theses and are in collaboration with Prof. Dr. Felix Otto.

15.1.2013
11:00 Uhr
Prof. Dr. Christoph Scheven
Universität Duisburg-Essen
Seminar: Localizable solutions to nonlinear parabolic problems with irregular obstacles SG 11

The talk is concerned with parabolic obstacle problems related to the evolutionary p-Laplace equation. The obstacle function may be very irregular and may also be increasing in time. In such a general situation, it was not known so far whether the solutions to such obstacle problems are localizable in the sense that the restriction of the solution on a subdomain again solves an obstacle problem. This is of course a serious problem if you are asking for local properties of a solution such as regularity. The talk will present a new existence result for localizable solutions to obstacle problems with irregular obstacles and will discuss some regularity properties of the constructed solutions.

15.1.2013
18:00 Uhr
Prof. Dr. Maria G. Westdickenberg
RWTH Aachen
Order and disorder: The competition between energy and entropy in randomly perturbed systems Hörsaal III

Abstract: Although at first glance a stochastic perturbation destroys the stability of energy minimizers, the probabilistic theory of large deviations reveals that the most likely pathways actually solve their own minimization problem. This theory—established in the mathe- matical literature decades ago with the classic book of Freidlin and Wentzell—treats stochastic ordinary differential equations. Subse- quently, it was confirmed that the theory generalizes naturally to stochastic partial differential equations. Recently, there has been progress in pushing large deviation theory to the limit and probing new regimes. In this talk, we will be particularly interested in the competition between energy and entropy that emerges in the case of small noise and large system size. We will present some recent results and, if time permits, give a feeling for the underlying tools. The talk is intended for a broad audience and no prior knowledge of probability theory will be assumed.

22.01.2013
11:00 Uhr
Prof. Dr. Günther Grün
Universität Erlangen-Nürnberg
Seminar: On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities SG 11

In this talk, we will be concerned with modeling and numerics of two-phase flow of immiscible, incompressible viscous fluids with different mass densities. In contrast to the case of identical mass densities, for general mass densities it has only recently been possible to formulate diffuse interface models which are consistent with thermodynamics and for which numerical schemes satisfying energy estimates can be developed. These models may be derived e.g. by Onsager's variational principle and they consist of a momentum equation for the velocity field coupled to a Cahn-Hilliard equation for the evolution of the order parameter. In the first part of the talk, we will discuss modeling aspects, and by a subtle discretization of the convective coupling between the flux of the phase-field and the momentum equation, we formulate a numerical scheme which is discretely consistent with thermodynamics. In the second part of the talk, we will prove its convergence in two and in three space dimensions and we present numerical simulations showing the full practicality of the approach.

18.12.2012
11:00 Uhr
Prof. Dr. Bernd Schmidt
Uni Augsburg
Seminar: On the passage from atomistic to continuum systems in elasticity and fracture mechanics SG 11

We study atomistic systems and their effective continuum counterpart in two different regimes: 1. Elastic deformations with given boundary data. Here our main aim is to relate discrete solutions of the Euler-Lagrange equations to solutions of the continuum equations of nonlinear elasticity. 2. crystals under uniaxial strain that may fracture. In this regime we will derive general cleavage laws from the atomistic interaction potentials.

27.11.2012
15:00 (special time)
Dr. Hans Knüpfer
Universität Bonn
Seminar: Well-posedness and lubrication approximation of the Darcy flow in the presence of a contact line SG 11

The propagation of a liquid drop on a plate is characterized by the evolution of the three-phase contact line where air, liquid and solid meet. The region occupied by the liquid is is described by a time-dependent domain with non-smooth boundary. We address well-posedness and regularity for models for the evolution of viscous fluids. We focus on certain models for fluid propagation of viscous liquids (Darcy flow, thin-film equation). Since the considered problems do not satisfy a maximum principle, the analysis is based on the dissipative structure of the models.

04.12.2012
14:00-15:30 Uhr
Postdoc Day
RWTH
Seminar: Barbora Benesova, Fabio Cavalletti, Michael Gelantalis, Marc Sedjro. Hauptgebäude 114

Barbora Benesova: Microstructures in solids and their mathematical treatment; Fabio Cavalletti: Optimal transportation in metric measure spaces; Michael Gelantalis: Rotating vortex solutions to the Gross-Pitaevskii equation on S^2; Marc Sedjro: On almost axisymmetric flows with forcing terms.

25.07.2012
15:00 Uhr
Ermal Feleqi
Seminar: Some aspects of Mean Field Games SG 11

Mean field games are a branch of game theory introduced by J-M. Lasry and P-L. Lions in order to model the behavior of a very large number of rational agents with a limited information (or visibility) of the game who optimize their decisions in view of the global (or macroscopic) informations available to them and that result from the actions of all agents. The perspective of applications is quite broad, e.g., in economics, finance, sociology, urban planning, engineering etc. Games with e very large number of players are approximated by a continuum limit'' (letting the number of the players go to infinity) in analogy with certain mean field'' approaches of statistical mechanics and physics, and this justifies the name. The focus on my talk will be on a class of ergodic stocastic differential games coupled only through the costs with players belonging to N different populations. (Each population consists of a large number of identical players, but the characteristics of the players vary from one population to the other.) In this case the MFG model results in a diagonal system of 2N stationary PDEs: N Hamilton-Jacobi-Bellman equations and N Kolmogorov-Fokker-Plank linear PDEs for the final distribution of the players of each population. I will exhibit a wide range of sufficient conditions for the solvability of these systems and their rigorous (or mathematical) derivation as a continuum limit' of certain systems of PDEs associated with games with a finite number of players as the cardinality of each population goes to infinity. In doing so, I do not only generalize previous work of Lasry and Lions by considering more general dynamics, costs and several populations, but also provide detailed proofs (which they do not in their articles). I will end my talk by indicating some perspectives for future research, most intriguing for me being the possibility of formulating a very general `master equation'' in metric spaces of probability measures. Its mathematical interpretation, study and degree of approximation of the related games with a finite number of players seem very fascinating and promising topics to me.

03.07.2012
11:00 Uhr
Barbora Benesova
Charles University Prague
Seminar: Young measures on invertible matrices SG 11
21.06.2012
11:00 Uhr
Xing Zhong
New Jersey Institute of Technology
Seminar: Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations SG 11

We study the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension. By using proper energy functional and exponentially weighted functional, for symmetric decreasing initial conditions, we can prove a one-to-one relation between the long time behavior of solution and the limit value of energy. Then, for a monotone increasing family of initial conditions, there exists a sharp transition between extinction and propagation. This is a joint work with Cyrill B. Muratov.

19.06.2012
11:00 Uhr
Dr. Greg Pavliotis
Imperial College
Seminar: Analysis and numerics for SPDEs with multiple scales SG 11

In this talk we will present analytical and numerical techniques for studying stochastic partial differential equations with multiple scales. After showing a rigorous homogenization theorem for SPDEs with quadratic nonlinearities, we present a numerical method for solving efficiently SPDEs with multiple scales. We then apply these analytical and numerical techniques to the stochastic Kuramoto-Shivashinsky equation for which we show that noise induced intermittent behavior and noise induced stabilization of solutions can occur. Finally, we show how ideas from parameter estimation for diffusion processes can be used in order to obtain low dimensional coarse-grained models from time series of the SPDE (projected onto the dominant modes). This is joint work with D. Blomker and M. Hairer (analysis), A. Abdulle (numerical analysis), M. Pradas Gene, D. Tseluiko, S. Kalliadasis, D.T. Papageorgiou (stochastic Kuramoto-Shivashinsky equation), S. Krmuscheid, S. Kalliadasis (data-driven derivation of coarse-grained models).

12.06.2012
11:00 Uhr
Dr. Michael Helmers
Uni Bonn
Seminar: Approximation of rotationally symmetric two-component bilayers SG 11

Models for two-component lipid bilayers assign an energy to a membrane that consists of an elastic part for each component and an interface term penalising component boundaries. Phase field approaches have been applied successfully to analyse such models, although many results are formal or rely on strong regularity assumptions. In the talk, we consider a surface phase field approximation of the spontaneous curvature model and study convergence to a sharp interface limit in a rotationally symmetric setting. Particular attention is paid to the regularity of limit membranes across component interfaces: exploring two different couplings between phase fields and elastic terms, we obtain either the classical model or an extension that assigns a "bending energy" also to possibly non-smooth connections.

05.06.2012
11:00 Uhr
Michael Gelantalis
Indiana University
Seminar: Rotating vortex solutions to the Gross-Pitaevskii equation on the two-sphere SG 11

In this talk we present some recent results concerning the existence of rotating vortex solutions to the Gross-Pitaevskii equation on S2.  The vortices that these solutions possess follow the trajectories of certain “relative equilibria” of the point-vortex problem for all time. Similar results have been rigorously established for the Gross-Pitaevskii equation posed on the flat two-torus and on bounded planar domains. We adopt an approach that is based on minimizing the Ginzburg-Landau energy subject to an integral momentum constraint within certain symmetry classes. We also discuss the orbital stability of these solutions within a class of symmetric initial data.

22.05.2012
11:00 Uhr
Dr. Matthias Kurzke
Uni Bonn
Seminar: The hydrodynamic limit of the parabolic Ginzburg-Landau equation SG 11

The Ginzburg-Landau functional serves as a model for the formation of vortices in various physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a point vortex system. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations with vortex sheet initial data. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. This is joint work with Daniel Spirn (University of Minnesota).

24.04.2012
11:00 Uhr
Dr. Luca Mugnai
MPI Leipzig
Seminar: Upper bounds on coarsening rates for surface attachment limited kinetics SG 11

Following an approach introduced by Kohn and Otto in [Comm. Math.Phys., 2002], we prove a weak form of an upper bound on the coarsening rate for surface attachment limited kinetics which is modeled by volume preserving mean curvature flow. (This is joint work with Christian Seis, Univ. of Toronto).

19.12.2011
ganztägig
3-Städte Seminar zur Analysis
AC-Dortmund-Eindhoven
TBA
14.11.2011
18:00 Uhr
Prof. Dr. Joost Hulshof,
VU Amsterdam
Kolloquium: Holefilling solutions of the porous medium equation HS III

Solutions of the porous medium equation have the finite speed of propagation property, exhibited by the source type solutions, which can be used as subsolutions to push the support of nontrivial nonnegative solutions outwards and force holes in the support to disappear in finite time. I will give an overview of what we know about how these holes disappear, with some emphasis on an instability result proved with Angenent, Aronson and Van den Berg.

08.11.2011
11.00 Uhr
Dr.Hendrik Weber,
University of Warwick
Seminar: Stochastic Allen-Cahn equation and Mean curvature flow SG 11

The Allen-Cahn equation is a classical model to describe the evolution of a system with two stable phases. It is well-known that in the sharp interface limit it approximates an evolution of phase indicator functions the boundary of which performs a motion by mean curvature. In this talk we investigate the stability of this result under perturbation with a random forcing. The difficulty lies in the low regularity of this noise term that makes the application of standard analytical methods difficult. We fist discuss a simple setting where the forcing is constant in space and a smoothened white noise in time. We show that in some diagonal limit where the noise smoothing goes to zero as the interface width goes to zero the solutions converge to a forced mean curvature flow. This is shown in the framework of classical solutions via an explicit construciton of sub- and supersolutions. Then we discuss the case of a stochastic transport term that is white in time and coloured in space. In this case we prove a priori bounds that imply compactness and regularity of the limiting evolution. We explain the difficulty of describing the limit.

11.01.2011
Prof. Dr. Ben Schweizer,
TU Dortmund
Kolloquium: Das Nadel-Problem: eine neue Methode für Homogenisierung HS III
19.10.2010,
14.00 Uhr
André Schlichting,
MPI Leipzig
Ein stochastisches Becker-Döring Modell - Pfadweise Betrachtung von Kolloidwachstum SG 11

Wir betrachten ein Modell für Clusterwachstum, in dem die Dynamik nur durch Interaktion zwischen Monomeren und Clustern (bestehend aus mehreren Monomeren) gegeben ist.
Bereits 1935 berechneten die beiden Physiker R. Becker und W. Döring mit Hilfe dieses einfachen Modells Nukleationsraten, welche sehr gut mit experimentellen Befunden übereinstimmten. Erste mathematische Untersuchungen des deterministischen Systems erfolgten in den 80er Jahren.
In neueren Ergebnissen wurde gezeigt, dass die LSW-Gleichungen (Lifshitz, Slyozov, Wagner) ein möglicher Grenzfall des Becker-Döring-Modells sind, welche den Prozess der Ostwald-Reifung beschreiben. Interpretiert man die gegebenen Übergangsraten stochastisch, so lassen sich auch Aussagen über Pfade einzelner Cluster machen. Anhand numerischer Experimente werden in dem Vortrag einige Fragestellung verdeutlicht und mögliche Lösungsansätze dargestellt.

22.07.2010,
18.00 Uhr
Prof. Dr. Jan Kristensen,
University of Oxford
On the problem of regularity in the calculus of Variations HS IV

There are by now several examples of regular variational problems that admit singular minimizers (non-differentiable Sobolev maps, and worse) in the multi-dimensional vectorial case. This is in line with what one can prove: minimizers are partially regular, meaning they are smooth outside a small relatively closed subset of their domain. We refer to this set of singularities as the singular set. General measure theory easily gives that the singular set has zero Lebesgue measure, but over the years much better estimates of their size, in particular in terms of Hausdorff measures, have been derived. Still the gap between the examples provided by singular minimizers and the theoretical bounds for the size of the singular sets remains a major challenge. In this talk we give a survey of recent results on size estimanes of singular sets.

14.07.2010,
18.15 Uhr
Prof. Dr. Barbara Niethammer,
University of Oxford
Old and new in the analysis of Ostwald ripening HS III

Ostwald ripening is a fundamental process in the aging of materials, where particles of a second phase embedded in a background phase interact by diffusional mass exchange to reduce their total surface area. In the low volume fraction regime the statistics of the ripening process have been described by the classical mean-field theory by Lifshitz, Slyozov and Wagner (LSW). However, due to several shortcomings of this theory, there has been considerable interest in taking higher order effects such as fluctuations in particle densities or collision of particles into account. Several models have been discussed in the applied literature, but existing theories are still partially contradicting. I will review recent progress in the mathematical analysis of Ostwald ripening shedding light on the consistent modeling of different higher order effects and their relevance respectively.

10.06.2010,
15.15 Uhr
Dr. Elena Villa,
University of Milan
On the outer Minkowski content of sets. Results and applications SG 11

Let E be a Borel set in Rd and Vol(E) be its volume; denoting by Er:={x ∈ Rd: dist(x,E) ≤ r} the parallel set of E at distance r, the limit of Vol(Er \ E)/r for r which goes to zero, is called outer Minkowski content of E, provided that the limit exists finite. We give general conditions, stable under finite unions, ensuring the existence of the outer Minkowski content of Borel subsets of Rd. In particular we show how the value of the outer Mikowski content turns out to be closely related to the density of E at its boundary points, and how our results also apply to the study of the differentiability of the volume function of bounded sets, extending some known results in literature. In particular, we provide sufficient conditions which imply the equality between the outer Minkowski content of E and its surface measure. Such an equality is of interest in image analysis and stochastic geometry problems (estimation of mean surface densities, and evolution equations of stochastic birth-and-growth processes, for instance); some applications will be discussed in the final part of the talk.

04.05.2010,
15.45 Uhr
Albert Nana,
LuF Mathematik
Oberseminar: Weak solutions for fourth-order parabolic equations modeling epitaxial thin film growth SG 11

The aim of the present work is to establish in an appropriate function space the existence solutions of the following nonlinear parabolic problem (0.1):

ut + Δ2u = ∇ ⋅ f(∇u) in Ω × (0,T)
Nu|∂Ω = ∂NΔu|∂Ω = 0
u|t=0 = u0

where Ω ⊂ ℜn is a bounded smooth domain, u0 is a initial data.
Throughout we assume the function f ∈ C1(Rn,Rn) to satisfy f(0)=0 and that there exist some α>0 such that |f'(λ)| ≤ C|λ|α for any λ ∈ Rn holds with C>0. A major reason for interest in (0.1) is that it modeles the epitaxial growth of nanoscale thin films, where u(x,t) denotes the height from the surface of the film in epixial growth. The term Δ2u denotes the capillarity-driven surface diffusion and ∇⋅f(∇u) the upward hopping of atoms. Our main tools are Lp-Lq estimate of linear part e2 and successives approximations.

18.03.2010,
14 Uhr
TU Eindhoven
Nonlinear micro-macro transmission conditions for two-scale reaction-diffusion systems: Modeling and analysis SG 11

We study a reaction-diffusion system posed at two different spatial scales which involves nonlinear reaction and gas-liquid mass-transfer terms. The system incorporates a non-standard component -- a nonlinear transfer function connecting micro and macro-transport of species including thus in the model nonlinear deviations from local equilibrium configurations [usually modeled in the literature, as a faute de mieux solution, by a linear expression -- the Henry's law]. We prove non-negativity and L^\infty-bounds for the active concentrations, uniqueness, and obtain global-in-time existence of weak solutions via a two-scale Galerkin method. For the case of 2D macroscopic and microscopic domains, we give an a priori bound on the rate of convergence of our two-scale Galerkin scheme. This research is done in collaboration with Maria-Neuss Radu (Universität Heidelberg) and Omar Lakkis (University of Sussex).

10.11.2009,
18 Uhr
Dr. Verena Bögelein,
Universität Parma
Kolloquium: Regularity of degenerate parabolic systems via the method of p-caloric approximation HS III

In this talk we are interested in the regularity of solutions to degenerate parabolic systems. A by now classical result due to DiBenedetto states that the spacial gradients of solutions to the parabolic p-Laplacean system are Hölder continuous. It is our aim to go somewhat beyond this result, in the sense that we consider degenerate parabolic systems of more general structure, i.e. not necessarily having the usual quasi-diagonal structure of Uhlenbeck type. Then, by counterexamples it is clear that everywhere regularity cannot hold in general. Nevertheless, we are able to establish a partial regularity result stating that solutions are regular almost everywhere, i.e. except for a set of Lebesgue measure zero.

The method of the proof is based on the p-caloric approximation lemma, which allows to approximate functions with solutions of the parabolic p-Laplacean system in a similar way as the classical harmonic approximation lemma (going back to DeGiorgi) does via harmonic functions.

21.07.2009,
18 Uhr
Dr. Roger Moser,
University of Bath
Kolloquium: Analysis aspects of polyharmonic maps HS III

A map between two Riemannian manifolds is called polyharmonic if it solves a certain type of variational problem. The simplest possible example is the Dirichlet energy for curves in a Riemannian manifold. Its critical points are geodesics, which are relatively easy to understand. When we study higher-dimensional domains and functionals involving higher derivatives, however, then the resulting variational problems can become challenging. This is due to nonlinearities, coming from the curvature of the target space, in the underlying differential operators and in the Euler-Lagrage equations. We discuss a few questions and results concerning existence and regularity of solutions.

09.06.2009,
16 Uhr
Nicolas Condette,
HU Berlin
A Fourier Collocation Method for a Pattern Forming Gradient Flow Equation: Analysis and Simulation. SG 11

We are interested in the morphology of patterns arising as local minimizers of a nonconvex and nonlocal functional that exhibits a competition of interfacial and dipolar energies. Typical applications include magnetic garnet films as well as diblock copolymers. Based on a relaxed energy functional, we implement and analyze a fully discrete Fourier collocation scheme in order to approximate solutions of the associated L^2 gradient flow equation. The latter reads as a nonlinear parabolic equation in 2D space. We prove existence and uniqueness of the numerical solution and show that it converges to a solution of the initial continuous problem. We also derive some error estimates and finally perform numerical experiments aimed at illustrating the theoretical results.

26.05.2009,
16 Uhr
Dr. Mariya Ptashnyk,
Universität Heidelberg
Derivation of a macroscopic model for diffusive transport of strongly-sorbed solute in the soil SG 11

One of the important questions in the soil science deals with the distribution of strongly-sorbed solutes in the soil. The experiments for strongly-sorbed ions, such as phosphate, show that ion bio-availability in the soil may be limited by their diffusion within the particles. A model for the diffusive transport of strongly-sorbed solutes in the inter-particle and intra-particle space is presented. The adsorption of solute on surfaces inside and outside soil particles is described by a nonlinear reaction. Using homogenization techniques effective macroscopic equations for the solute movement in the soil are derived. Applying the two-scale convergence, we show that the sequence of solutions of the original problem converges to the solution of the macroscopic problem. To show the convergence of the nonlinear terms on the surfaces the periodic modulation (unfolding method) is used. Solutions of the macroscopic model are compared with the data from phosphate pulse experiment.

17.12.2008,
16 Uhr
Dr. Kianhwa Colin Djie,
Lehrstuhl I für Mathematik
Blow-up in endlicher Zeit bei einem Modell der Chemotaxis unter Berücksichtigung des volumenfüllenden Effekts SG 11

Wir betrachten das folgende parabolisch-elliptische Keller-Segel-Modell der Chemotaxis

mit Anfangsdatum , worin den Mittelwert von bezeichnet. Durch Behandlung der Nichtlinearität berücksichtigt dieses System den volumenfüllenden Effekt im Sinne von Painter und Hillen (Can. Appl. Math. Q.10, 501-543 (2002)). Während die globale Existenz in diversen Varianten der Chemotaxis-Modelle bekannt ist, gibt es wenig Resultate über Blow-up in endlicher Zeit. In dem Vortrag wird eine auf dem Vergleichsprinzip beruhende Methode vorgestellt, mit der man im radialsymmetrischen Fall einen Blow-up in endlicher Zeit nachweisen kann für und bei Parametern , und in Raumdimension , sofern die Masse des Anfangswerts stark genug im Ursprung konzentriert ist. Diese Parameterwahl ist insofern optimal, als globale Existenz und Beschränktheit der Lösung für und im Fall gezeigt werden kann.