|Talks in the winter term 2019/2020|
|15.10.2019||Nick Lindemulder (Karlsruhe)||An Intersection Representation for a Class of Anisotropic Vector-valued Function Spaces|
In this talk we discuss an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting à la Hedberg & Netrusov, which includes weighted anisotropic mixed-norm Besov and Triebel-Lizorkin spaces. In the special case of the classical Triebel-Lizorkin spaces, the intersection representation gives an improvement of the well-known Fubini property. The motivation comes from the weighted -maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Triebel-Lizorkin spaces occur as spaces of boundary data.
|22.10.2019||Bas Nieraeth (Karlsruhe)||Weighted theory and extrapolation for multilinear operators|
|19.11.2019||Andreas Geyer-Schulz (Karlsruhe)||On global well-posedness of the Maxwell–Schrödinger system|
|02.12.2019||Wenqi Zhang (Canberra)||Localisation of eigenfunctions via an effective potential for Schrödinger operators|
For Schrödinger operators with potentials (possibly random) we introduce the Landscape function as an effective potential. Due to the nicer properties of this Landscape function we are able to recover localisation estimates for continuous potentials, and specialise these estimates to obtain an approximate diagonalisation. We give a brief sketch of these arguments.
This talk is taking place at 10:30 am in Seminar room 2.066.
|03.12.2019||Yonas Mesfun (Darmstadt)|| On the stability of a chemotaxis system with logistic growth|
In this talk we are concerned with the asymptotic behavior of the solution to a certain Neumann initial-boundary value problem which is a variant of the so-called Keller-Segel model describing chemotaxis. Chemotaxis is the directed movement of cells in response to an external chemical signal and plays an important role in various biochemical processes such as e.g. cancer growth.
We show a result due to Winkler which says that under specific conditions, there exists a unique classical solution to this Neumann problem which converges to the equilibrium solution with respect to the -norm. For this purpose we study the Neumann Laplacian in , in particular some decay properties of its semigroup and embedding properties of the domain of its fractional powers, and then use those properties to prove Winkler's result.
|10.12.2019||Emiel Lorist (Delft)|| Singular stochastic integral operators: The vector-valued and the mixed-norm approach|
Singular integral operators play a prominent role in harmonic analysis. By replacing integration with respect to some measure by integration with respect to Brownian motion, one obtains stochastic singular integral operators, which arise naturally in questions related to stochastic PDEs. In this talk I will introduce Calderón-Zygmund theory for these singular stochastic integral operators from both a vector-valued and a mixed-norm viewpoint.
|14.01.2020||Alex Amenta (Bonn)|| Vector-valued time-frequency analysis and the bilinear Hilbert transform|
The bilinear Hilbert transform is a bilinear singular integral operator (or Fourier multiplier) which is invariant not only under translations and dilations, but also under modulations. This additional symmetry turns out to make proving -bounds especially difficult. I will give an overview of how time-frequency analysis is used in proving these -bounds, with focus on the recently understood setting of functions valued in UMD Banach spaces.
|21.01.2020||Willem van Zuijlen (Berlin)|| Spectral asymptotics of the Anderson Hamiltonian|
In this talk I will discuss the asymptotics of the eigenvalues of the Anderson Hamiltonian, which is the operator given by . We consider to be (a realisation of) white noise and consider the operator on a box with Dirichlet boundary conditions. I will discuss the result in joint work with Khalil Chouk: almost surely the eigenvalues divided by the logarithm of the size of the box converge to the same limit. I will also discuss the application of this to obtain the large-time asymptotics of the total mass of the parabolic Anderson model, which is the SPDE given by .
|18.02.2020||TULKKA in Konstanz|
The talks take place in Room A 704 (University of Konstanz).
|11:45-12:15||Adrian Spener (Ulm)||Curvature-dimension inequalities for nonlocal operators|
|13:45-14:30||Sophia Rau (Konstanz)||Stability results for thermoelastic plate-membrane systems|
|14:45-15:30||Andreas Geyer-Schulz (Karlsruhe)||On global well-posedness of the Maxwell-Schrödinger system|
|16:15-17:00||Delio Mugnolo (Hagen)||Linear hyperbolic systems|
|Talks in the summer term 2019|
|21.05.2019||Lucrezia Cossetti (Karlsruhe)||Multipliers method for Spectral Theory.|
Originally arisen to understand characterizing properties connected with dispersive phenomena, in the last decades the multipliers method has been recognized as a useful tool in Spectral Theory, in particular in connection with proof of absence of point spectrum for both self-adjoint and non self-adjoint operators.
Starting from recovering very well known facts about the spectrum of the free Laplacian in , we will see the developments of the method reviewing some recent results concerning self-adjoint and non self-adjoint perturbations of this Hamiltonian in different settings, specifically both when the configuration space is the whole Euclidean space and when we restrict to domains with boundary. We will show how this technique allows to detect physically natural repulsive and smallness conditions on the potentials which guarantee the absence of eigenvalues. Some very recent results concerning Pauli and Dirac operators will be presented too.
The talk is based on joint works with L. Fanelli and D. Krejcirik.
|28.05.2019||Philipp Harms (Freiburg)||Smoothness of the functional calculus and applications to variational PDEs.|
The functional calculus, which maps operators to functionals , is holomorphic for a certain class of operators and holomorphic functions . In particular, fractional Laplacians depend real analytically on the underlying Riemannian metric in suitable Sobolev topologies. As an application, this can be used to prove local well-posedness of some geometric PDEs, which arise as geodesic equations of fractional order Sobolev metrics.
Joint work with Martins Bruveris, Martin Bauer, and Peter W. Michor.
|30.07.2019||TULKKA in Karlsruhe|
Die Vorträge finden in Raum 1.067 im Kollegiengebäude Mathematik (20.30) statt.
|11:30-12:15||Lucrezia Cossetti (Karlsruhe)|| Unique continuation for the Zakharov-Kuznetsov equation |
In this talk we analyze uniqueness properties of solutions to the (2+1)-Zakharov-Kuznetsov (ZK) equation
Mainly motivated by the very well known PDE's counterpart of the Hardy uncertainty principle, we provide a two times unique continuation result. More precisely, we prove that given two solutions to ZK, as soon as the difference decays (spatially) fast enough at two different instants of time, then As expected, it turns out that the decay rate needed to get uniqueness reflects the asymptotic behavior of the fundamental solution of the associated linear problem. Encouraged by this fact we also prove optimality of the result.
Some recent results concerning the (3+1)- dimensional ZK equation will be also presented.
The seminar is based on a recent paper (CFL) in collaboration with L. Fanelli and F. Linares.
(CFL) L.Cossetti, L.Fanelli and F.Linares, Uniqueness results for Zakharov-Kuznetsov equation, Comm. Partial Differential Equations, DOI:10.1080/03605302.2019.1581803
|14:00-14:45||Marius Müller (Ulm)|| The biharmonic Alt-Caffarelli problem |
We discuss a variational free boundary problem of Alt-Caffarelli type. We consider the energy
defined for a membrane that is spiked at some positive level on . The two summands impose conflicting interests on minimizers: Little bending versus a large region of nonpositivity. We study regularity of minimizers and the free boundary, which happens to be the nodal set .
As it will turn out, each minimizer has non-vanishing gradient on its free boundary, which connects the regularity of the two objects. Regularity discussion of the minimizer leads to the study of measure-valued Dirichlet problems and carries a potential theoretic flavor.
|15:00-15:45||Karsten Herth (Konstanz)|| Decay rates for anisotropic Reissner-Mindlin plates |
The Reissner-Mindlin plate is a model for thick plates, where the mid-surface normal is not required to remain perpendicular to the mid-surface. We analyze the asymptotic behavior of anisotropic thermoelastic Reissner-Mindlin plate equations in the whole space, using the Fourier transform and the method of stationary phase. This leads to Fresnel-like surfaces, similar to those in anisotropic elasticity, whose points of vanishing curvature are linked with the decay behavior.
|16:00-16:45||Kaffeepause (in Raum 1.058)|
|16:45-17:30||Lahcen Maniar (Marrakesch)|| Null controllability for a heat equation with dynamic boundary condition and drift terms |
We consider the heat equation in a space bounded domain subject to dynamic boundary conditions of surface diffusion type and involving drift terms in the bulk and in the boundary. We prove that the system is null controllable at any time. The results is based on new Carleman estimates for these type of boundary conditions. We conclude by new results on semilinear equations with non linear functions occuring a blow up of the solutions without control.
|ab 18:00||Abendessen im Restaurant "Il Caminetto" (Kronenstr. 5)|
|Talks in the winter term 2018/2019|
|29.01.2019||Fabian Hornung (Karlsruhe)||Neural Network Approximation for high dimensional Kolmogorov PDE.|
First we introduce some basic concepts in the theory of artificial neural networks (ANNs) and present how Kolmogorov PDEs can be reformulated as a minimization problem using techniques from stochastic analysis. This can serve as foundation of deep learning algorithms to numerically solve these PDEs.
In the second part of the talk, we sketch a proof that artificial neural networks approximate the PDE-solution without curse of dimensionality, i.e. that the number of parameters of the approximating ANN is bounded by a polynomial in the dimension of the problem and the reciprocal of the accuracy.
The talk is based on joint work with Philipp Grohs, Arnulf Jentzen, and Philippe von Wurstemberger.
The talk takes place at 15:45 in room 3.060.
|19.02.2019||TULKKA in Ulm|
|11:30-12:15||Lisa Fischer (Konstanz)||Generalized thermoelastic plate: Well-posedness and frequency analysis|
|13:45-14:30||Konstantin Zerulla (Karlsruhe)||Ein ADI-Verfahren mit gleichmäßig exponentiell stabilen Approximationen für die Maxwell-Gleichungen|
|14:45-15:30||Emil Wiedemann (Ulm)||Analysis of Turbulent Flows: Compressible and Incompressible|
|16:15-17:00||Patrick Tolksdorf (Darmstadt)||A smooth introduction to fluid mechanics in rough domains|
|ab 17:45||Dinner in the city center of Ulm|
|More information on Tulkka can be found here.|
|12.03.2019||Boris Baeumer (Dunedin)||Boundary conditions for Levy processes on bounded domains and their governing PDEs.|
Levy processes are jump processes governed by non-local operators on and are used to model dispersive systems where the occasional large dispersal event (many standard deviations) is driving the system. In modelling, boundaries appear naturally and in 1D we answer the question of what type of boundary condition for the non-local operator corresponds to what type of boundary behaviour of the process by using numerical approximation schemes.
The talk takes place in room 2.066 from 14:00 to 15:00.
|12.03.2019||Petru Cioica-Licht (Essen)||SPDEs on domains with corner singularities.|
Although there exists an almost fully-fledged -theory for (semi-)linear second order stochastic partial differential equations (SPDEs, for short) on smooth domains, very little is known about the regularity of these equations on non-smooth domains with corner singularities. As it is already known from the deterministic theory, corner singularities may have a negative effect on the regularity of the solution. For stochastic equations, this effect comes on top of the already known incompatibility of noise and boundary condition. In this talk I will show how a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary may be used in order to deal with both sources of singularity and their interplay.
The talk takes place in room 2.066 from 15:00 to 16:00.
|Talks in the summer term 2018|
|09.05.2018||Pierre Portal (Canberra)||An operator theoretic generalisation of pseudo-differential calculus.|
The calculus is an operator theoretic construction that allows one to extend Fourier multiplier theory, towards rough settings in particular. In this talk, we consider a similar functional calculus that extends pseudo-differential operator theory. It involves two group generators satisfying the canonical commutator relations, and thus generalising the usual position and momentum operators. I'll discuss a transference result relating this calculus to twisted convolutions on Bochner spaces, and a formula connecting it to the calculus of abstract harmonic oscillators. The latter allows us, in particular, to show using Kriegler-Weis theory that these harmonic oscillators have a Hormander calculus.
The talk takes place at 10:00 in room 2.067.
|17.07.2018||TULKKA in Konstanz|
|11:45-12:30||Marcel Kreuter (Ulm)||Vektorwertige elliptische Randwertprobleme auf rauen Gebieten|
|13:45-14:30||Martin Spitz (Karlsruhe)||Lokale Wohlgestelltheit nichtlinearer Maxwell-Gleichungen mit perfekt leitenden Randbedingungen|
|14:45-15:30||Sita Siewert (Tübingen)||Exponentielle Dichotomie und Spektrum dynamischer Banach-Moduln|
|16:15-17:00||Gieri Simonett (Nashville)||On the Muskat problem|
|ab 17:45||Dinner in the restaurant "Hafenhalle"|
|More information on Tulkka can be found here.|
|Talks in the winter term 2017/2018|
|21.11.2017||Konstantin Zerulla (Karlsruhe)||Stabilitätserhaltende diskretisierte Approximationen gedämpfter Wellengleichungen.|
|28.11.2017||Martin Spitz (Karlsruhe)||Nichtlineare Maxwellgleichungen - Blow-up Kriterium und stetige Abhängigkeit.|
|05.12.2017||Yuri Tomilov (IM PAN, Warsaw)||Why do circles in the spectrum matter?|
We present several results linking the joint numerical ranges of Hilbert space operator tuples to the circle structure of the spectrum of tuples. We will explain how our approach allows us to unify, extend or supplement several results where the circular structure of the spectrum is crucial: Arveson's theorem on almost-wandering vectors of unitary actions, Brown-Chevreau-Pearcy's theorem on invariant subspaces of Hilbert space contractions and Hamdan's recent result on supports of Rajchman measures, to mention a few. Moreover, we will give several applications of the approach to new operator-theoretical constructions inverse in a sense to classical power dilations.
This is joint work with V. Müller (Prague).
|30.01.2018||Luca Hornung (Karlsruhe)||Wohlgestelltheit einer nichtlinearen Maxwell-Gleichung mit retardiertem Materialgesetz.|
|06.02.2018||TULKKA in Karlsruhe|
The talks take place in room 1.067 in the math building.
|11:30-12:15||Marie-Luise Hein (Ulm)|| Das Prinzip der linearisierten Stabilität für parabolische Volterra Gleichungen|
In diesem Vortrag werde ich das Prinzip der linearisierten Stabilität für parabolische Volterra Gleichungen für den Spezialfall des Standard-Kerns erläutern. Anschließend werde ich ein Stabilitätsresultat für quasilineare zeit-fraktionelle Evolutionsgleichungen in der Situation der maximalen -Regularität präsentieren.
|14:00-14:45||Tim Binz (Tübingen)|| Gleichmäßig elliptische Operatoren mit Wentzell Randbedingung und der Dirichlet-zu-Neumann Operator|
Für gleichmäßig elliptische Operatoren und die zugehörigen konormalen Ableitungen auf stetigen Funktionen gelang es J. Escher 1994 zu zeigen, dass der assoziierte Dirichlet-zu-Neumann Operator Generator einer analytischen Halbgruppe ist.
Im ersten Teil des Vortrages werden wir den Zusammenhang zwischen Operatoren mit Wentzell Randbedingungen und Dirichlet-zu-Neumann Operatoren studieren.
Dazu wird ein abstrakter Rahmen eingeführt, der eine Diskussion von Randwertproblemen mit Hinblick auf Generatoreneigenschaften erlaubt. Anschließend geben wir einen alternativen Beweis für die Aussage von Escher, der es uns zusätzlich erlaubt den Winkel zu berechnen. Darüber hinaus verallgemeinern wir die Aussage auf kompakte Mannigfaltigkeiten mit Rand.
|15:00-15:45||Felix Kammerlander (Konstanz)|| Exponentielle Stabilität für ein gekoppeltes System von ungedämpft-gedämpften Plattengleichungen|
Wir betrachten ein Transmissionsproblem elastischer Platten in einem Gebiet , welches aus einem inneren Gebiet mit und einem äußeren Gebiet , welches umschließt, besteht. In betrachten wir eine ungedämpfte Plattengleichung, in hingegen eine strukturell gedämpfte Platte. Mithilfe passender Transmissionsbedingungen sind die beiden Gleichungen an der Grenzschicht der beiden Gebiete miteinander gekoppelt.
Unter Verwendung von Halbgruppentheorie zeigt man die Wohlgestelltheit des Problems in einem geeigneten Hilbertraum. Die Energie des Gesamtsystems nimmt ab, wobei der Verlust der Energie einzig und allein durch die Dämpfung in verursacht wird. Wir zeigen, dass die Dämpfung in bereits stark genug ist, um exponentielles Abklingen der Energie für das Gesamtsystem zu erhalten. Hierfür wird eine gewisse a-priori Abschätzung für das parabolische System der gedämpften Platte benötigt.
|16:45-17:30||Amru Hussein (Darmstadt)|| Beyond maximal -regularity - a case study in spaces of bounded functions|
For semilinear equations the maximal -regularity approach gives local well-posedness for initial values in trace spaces. For typical second order parabolic problems these lie between the ground space and . In particular some differentiability is necessary.
In particular cases one can weaken the assumptions to consider rough initial data without differentiability assumptions by moving to the end point of the scale, i.e. considering . Here, we illustrate this for the case of the primitive equations. This is a geophysical model derived from Navier-Stokes equations assuming a hydrostatic balance. We prove that the combination of heat semigroup and Riesz transforms is a bounded operator in spaces of bounded functions and that this combination satisfies certain smoothing properties. This is essential to tackle the semilinear problem by an evolution equation approach. The classical maximal -regularity approach gives additional regularity properties, and suitable a priori bounds lead to a global solution even for rough initial data.
|ab 18:00||Dinner in the restaurant "Il Caminetto" (Kronenstr. 5)|
|More information on Tulkka can be found here.|
|21.02.2018||Emiel Lorist (Delft)||Vector-valued extrapolation to Banach function spaces.|
If an operator is bounded on for some and all weights in the class of Muckenhoupt weights , then extends to a bounded operator on the Bochner space for any Banach function space with the UMD property, which is a vector-valued extrapolation theorem by Rubio de Francia. In this talk I will discuss several generalizations of this theorem. In particular I will present a multilinear limited range version for vector-valued extrapolation to Banach function spaces and discuss various applications, including vector-valued Littlewood-Paley-Rubio de Francia-type estimates, the -boundedness of Fourier multipliers and the variational Carleson operator, and boundedness of the vector-valued bilinear Hilbert transform.
This is joint work with Alex Amenta, Bas Nieraeth and Mark Veraar (TU Delft).
See the german page for older talks.