|Talks in summer 2022|
|03.05.2022, 2:00 pm||Peer Christian Kunstmann (Karlsruhe)|
Functional calculi for Stokes operators with first order boundary conditions on unbounded domains
We study functional calculi in for Stokes operators with Hodge, Navier, and Robin type boundary conditions on uniform -domains . Our research complements recent results on the -theory of such operators and also sheds new light on the cases and .
|17.05.2022, 2:00 pm||Dorothee Frey (Karlsruhe)|
Strichartz and dispersive estimates for equations with structured Lipschitz coefficients
We shall discuss Strichartz estimates for both Schrödinger and wave equations with structured Lipschitz coefficients. The arguments are based on Phillips calculus, which allows to deduce dispersive estimates from the constant coefficient case. For fixed time estimates we require a more refined wave packet analysis.
|24.05.2022, 2:00 pm||Christopher Bresch (Karlsruhe)|
Local wellposedness of Maxwell systems with scalar-type retarded material laws
In the first part of the talk, local wellposedness of an abstract retarded evolution equation is studied using the concept of a mild solution and Banach's fixed point theorem. The second part is an application to Maxwell equations in the context of a model from nonlinear optics.
|31.05.2022, 2:00 pm||Robert Schippa (Karlsruhe)|
Strichartz estimates for Maxwell equations on domains with perfectly conducting boundary conditions
We consider Maxwell equations on a domain with perfectly conducting boundary conditions in isotropic media. In the charge-free case we recover Strichartz estimates due to Blair-Smith-Sogge for wave equations on domains. We shall also consider the quasilinear case of the Kerr nonlinearity, in which case we recover the Strichartz estimates and well-posedness results from Euclidean space. This is joint work with Nicolas Burq (Universite Paris-Sud).
|14.06.2022, 2:00 pm||Martin Spitz (Bielefeld)|
Almost sure scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data
The local and global wellposedness theory of nonlinear dispersive equations with randomized data has attracted a lot of interest over the last years. In particular in the scaling-supercritical regime, where a deterministic wellposedness theory fails, randomization has become an important tool to study the generic behaviour of solutions.
In this talk we study the energy-critical NLS on with supercritical initial data. We present a randomization based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and a unit-scale decomposition in physical space. We then discuss the resulting (almost surely) improved space-time estimates for solutions of the linear Schrödinger equation with randomized data and how these estimates yield almost sure scattering for the energy-critical cubic NLS.
|12.07.2022, 2:00 pm||Richard Nutt (Karlsruhe)|
Exponential Decay of the quasilinear Maxwell system due to surface conductivity
We consider a quasilinear Maxwell system on a bounded domain with absorbing boundary conditions and derive an estimate for the normal traces of the electric and magnetic field. This allows us to strengthen a result on exponential decay for small initial data.
|25.07.2022||TULKKA in Ulm|
|11:30-12:15||Robert Schippa (Karlsruhe)||Quasilinear and time-harmonic Maxwell equations|
|13:45-14:30||David Ploß (Konstanz)||The Bi-Laplacian with Wentzell boundary conditions on Lipschitz Domains|
|14:40-15:25||Dennis Gallenmüller (Ulm)||Measure-valued low Mach limits|
|16:10-16:55||Angkana Rüland (Heidelberg)||On Rigidity, Flexibility and Scaling Laws: The Tartar Square|
|ab 17:45||Dinner in the city centre of Ulm|
|Detailed information on TULKKA is available hier.|
|26.07.2022, 2:00 pm||Maximilian Ruff (Karlsruhe)|
A Fourier Integrator for the cubic Schrödinger equation at regularity
In this talk I will present the topic of my master thesis which is based on a paper by Ostermann, Rousset and Schratz. They introduced a new exponential-type time integration method for the cubic Schrödinger equation. While only assuming regularity of the solution, -Convergence with order strictly larger than 1/2 was shown. An important ingredient in this context are discrete-time Strichartz estimates.
|Talks in winter 2021/2022|
|16.11.2021, 2:00 pm||Robert Schippa (Karlsruhe)|
Resolvent estimates for time-harmonic Maxwell's equations
We prove resolvent estimates for time-harmonic Maxwell's equations in -spaces with pointwise, spatially homogeneous, and possibly anisotropic material laws. These allow for the proofs of Limiting Absorption Principles and construction of solutions. In the fully anisotropic case, which is joint work with Rainer Mandel, the construction relies on new Bochner-Riesz estimates with negative index for non-elliptic surfaces.
arXiv:2103.16951: Resolvent estimates for time-harmonic Maxwell's equations in the partially anisotropic case
arXiv:2103.17176: Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces
|30.11.2021, 2:00 pm||Konstantin Zerulla (Karlsruhe)|
Time integration of Maxwell equations with low regularity
The Peaceman-Rachford alternating direction implicit (ADI) scheme is very attractive for the time discretization of linear Maxwell equations on cuboids. Indeed, it is numerically stable and has optimal linear complexity. During the error analysis, it however turns out that the accuracy of the scheme heavily depends on the regularity of the solution of the Maxwell equations.
In the talk, we study linear time-dependent Maxwell equations on a heterogeneous cuboid, modelling a rectangular waveguide. The medium is assumed to consist of several small subcuboids with different material properties. Here the solution of the Maxwell equations is not -regular. We hence construct a new directional splitting scheme for the time discretization, and we provide a rigorous time-discrete error estimate for the scheme in .
|03.12.2021, 10:00 am in Zoom||Jean-Claude Cuenin (Loughborough)|
Schrödinger operators with complex potentials
I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. We are interested in the magnitude and distribution of eigenvalues, and we seek bounds that only depend on an norm of the potential.
These questions are well understood for real potentials, but completely new phenomena arise for complex potentials. I will explain how techniques from harmonic analysis, particularly those related to Fourier restriction theory, can be used to prove upper and lower bounds. We will also discuss some open problems. The talk is based on recent joint work with Sabine Bögli (Durham).
|07.12.2021, 2:00 pm||Lucrezia Cossetti (Karlsruhe)|
Eigenvalue bounds and spectral stability of Lamé operator with complex potential
In this talk I will show how to get quantitative bounds on the location of eigenvalues, both discrete and embedded, of the non-self-adjoint Lamé operator of elasticity in terms of suitable norms of the potential. In order to do that we will use a nowadays well-oiled machinery based on the use of the Birman-Schwinger principle together with suitable uniform resolvent estimates. To emphasise the challenging feature of problems involving non-self-adjoint operators, we will show how, in the self-adjoint framework, such spectral enclosures are easily obtained as a consequence of the variational characterisation of the spectrum (no-longer available in a complex-valued context) and Sobolev inequalities.
The talk is based on the following two works:
L. Cossetti, Bounds on eigenvalues for perturbed Lamé operators with complex potentials, Math. Eng. 4 (2021), 5, 1-29
B. Cassano, L. Cossetti, L. Fanelli, Eigenvalue bounds and spectral stability of Lamé operator with complex potentials, Journal of Differential Equations 298 (2021), 528-559
|21.12.2021, 2:00 pm||Marco Fraccaroli (Bonn)|
Duality for outer spaces and relation to tent spaces
The theory of spaces for outer measures, or outer spaces, was introduced by Do and Thiele to encode the proof of boundedness of certain multilinear operators in a streamlined argument. Therefore, it was developed in the direction of the real interpolation features, e.g. Hölder's inequality and Marcinkiewicz interpolation, while other questions remained untouched. For example, whether the outer quasi-norms are equivalent to norms and satisfy any reasonable (Köthe) duality property.
In this talk, we will answer these questions, with a particular focus on two settings in the upper half plane and in the upper half space that are relevant in harmonic analysis. This allows us to clarify the relation between the outer spaces and the tent spaces introduced by Coifman, Meyer and Stein, and to get a glimpse at the use of the outer machinery in the proof of boundedness of multilinear operators mentioned above.
|11.01.2022, 2:00 pm||Constantin Bilz (Birmingham)|
Large sets without Fourier restriction theorems
Fourier restriction inequalities associated to submanifolds form a major area of research in harmonic analysis and are connected to problems in many other fields. Around 25 years ago, Mockenhaupt and Mitsis showed that the restriction of the Fourier transform to a fractal (instead of a submanifold) can also behave interestingly. But the class of fractals is vast and necessary conditions for restriction inequalities are not well understood. In this direction, I will show that Fourier restriction sets avoid a certain universal set of full Hausdorff dimension.
|11.01.2022, 4:00 pm in Zoom||Paweł Plewa (Wrocław)|
Hardy's inequality associated with orthogonal expansions
Let be a measure space, where and is doubling, and choose an orthonormal basis in . Hardy’s inequality on , , associated with has the following form
where , the symbol stands for the inner product in , and denotes an appropriate Hardy space. Here is a positive number referred to as the admissible exponent.
I will discuss a method of proving Hardy’s inequality for a large class of orthogonal bases, which I established in my PhD thesis. It relies on study of the corresponding semigroup (usually heat or Poisson semigroup). The strength of this method is evidenced by its generality and sharpness. This means that it can be applied in most (if not all) of the classical systems (e.q. Hermite, Laguerre, Jacobi) and it leads to the smallest possible admissible exponent.
If there is time left, I will mention my other projects, both finished and ongoing.
|18.01.2022, 2:00 pm||Peer Christian Kunstmann (Karlsruhe)|
Continuity properties of semigroups in real interpolation spaces
Starting from a bi-continuous semigroup (which might actually be strongly continuous) in a Banach space we investigate continuity properties of the semigroup that is induced in real interpolation spaces between and the domain of the generator. Of particular interest is the case with second index infinity. We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms, and indicate applications.
|01.02.2022, 2:00 pm||Patrick Tolksdorf (Mainz)|
On off-diagonal behavior of the generalized Stokes operator
Let denote a second-order elliptic operator in divergence form and let denote the corresponding strongly continuous heat semigroup on . If denote measurable sets with and if is supported in , then by the strong continuity of the semigroup, one finds that
An estimate that quantifies the convergence rate is often viewed as an off-diagonal estimate and it is well-known, that heat semigroups satisfy the following type of off-diagonal decay
In this talk, we study off-diagonal behavior of the generalized Stokes semigroup that is generated on by the generalized Stokes operator with bounded measurable coefficients , formally given by
In contrast to the elliptic operator , the operator exhibits a non-local behavior due to the presence of the pressure function . This non-locality affects the non-local behavior of the generalized Stokes semigroup and it is not clear how fast the support of a divergence free vector field that is supported in a set is smeared out. In this talk, first results in this direction are presented. We further discuss how possible optimal estimates could look like and try to pinpoint what has to be improved in the existing proof.
|Talks in the summer term 2021|
The talks are streamed via ZOOM.
|May 11, 2021, 2 pm||Dr. Michela Egidi (Bochum)|
The Control Problem for the heat equation on rectangular domains
We study the internal controllability of the heat equation on unbounded rectangular domains, for example the whole space, the half-space or the infinite strip, where the control set is a thick subset, meaning that it is well-distributed in the domain. We show that thickness is a sufficient condition for null-controllability of the system and that it is also possible to provide a sharp bound for the control cost in terms of the geometric parameters of the problem. We will also show that thickness is necessary for null-controllability.
|Talks in the winter term 2020/2021|
|23.02.2021, 11:00 am||Alexander Wittenstein (Kiel)|
Local Hardy spaces on measured metric spaces with local doubling property
After a short recap of the euclidean situation, we will introduce local Hardy spaces for and on more general measured metric spaces satisfying only some mild geometric assumptions. It will turn out, similar to the euclidean case, that the dual space of is the space of functions with (local) bounded mean oscillation to the parameters and , which we will denote by . We will then show, that both the local Hardy spaces and the spaces are independent of the parameters and , and will therefore all be denoted by and respectively. We will then conclude, that for the function space is a complex interpolation space between and . Therefore, we get by duality, that for the space is an interpolation space between and . With that in mind, we will finally give some criteria under which linear Operators are bounded from to .
|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.