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Archive of the Research Seminar

Talks in winter 2022/23

04.10.2022, 2:00 pm	Daniel Böhme (Mainz)

Über Operator adaptierte Hardy Räume
In 2005 haben Duong et al. zum ersten Mal Operator adaptierte Hardy Räume eingeführt, um die schönen Eigenschaften, die die klassischen Hardy Räume mit dem Laplace Operator verbinden, auf allgemeinere Operatoren zu übertragen. In diesem Vortrag betrachten wir sektorielle Operatoren $L$ auf $L^2(X)$ , wobei $X$ ein Raum von homogenen Typ ist, die einen beschränkten holomorphen Funktionalkalkül auf $L^2(X)$ haben und deren Wärmeleitungshalbgruppe, $(e^{−tL})_{t>0}$ , Davies-Gaffney Abschätzungen erfüllt. Nach einem kurzen Ausflug in die von Coifman, Meyer und Stein eingeführten Tent spaces, werden wir uns den Operator adaptierten Hardy Räumen $H^p_L(X)$ zuwenden. Für $0 < p \le 1$ definieren wir diese dann auf verschiedene Weisen und werden sehen, dass die verschiedenen Definitionen äquivalent sind. Ähnlich zu den klassischen Hardy Räumen werden wir zuletzt zeigen, dass die zu $L$ assoziierte Riesz Transformation beschränkt von $H^p_L(X)$ nach $L^p(X)$ abbildet.

15.11.2022, 2:00 pm in Zoom	Robert Schippa (Karlsruhe)

Quasilinear Maxwell Equations with Anisotropic Material Laws
We consider Maxwell equations with pointwise, fully anisotropic material laws. The characteristic surface is given by the Fresnel surface, which contains conical singularities. We prove Strichartz estimates for Hölder-continuous coefficients, which allows us to solve quasilinear Maxwell equations in the fully anisotropic case for rough initial data.

20.12.2022, 2:00 pm	Constantin Bilz (Karlsruhe)

An invitation to the regularity of maximal functions
The Hardy–Littlewood maximal inequality bounds maximal functions on Lebesgue spaces, but it does not reveal much about the behaviour of their gradients. Following a 1997 paper of Kinnunen, an area of research has developed that is concerned with the behaviour of maximal functions on Sobolev spaces. In this talk, we give an overview of this theory and use plenty of examples to illustrate its key features. We also present some recent results and open questions.

31.01.2023, 2:00 pm	Yonas Mesfun (Karlsruhe)

Strichartz estimates for Wave equations with structured coefficients

14.02.2023		TULKKA in Karlsruhe (Seminar room 1.067)

11:30-12:15	Constantin Bilz (Karlsruhe)	Large sets without Fourier restriction theorems
12:15-14:00		Lunch
14:00-14:45	Alexandra Neamtu (Konstanz)	A semigroup approach to quasilinear rough PDEs
15:00-15:45	Lukas Niebel (Ulm)	Kinetic maximal $L^p_\mu$ -regularity
16:00-16:45		Coffee Break
16:45-17:30	Moritz Egert (Darmstadt)	Four Critical Numbers for Elliptic Systems with Block Structure
from 18:00		Dinner in the restaurant "Il Caminetto" (Kronenstr. 5)
		More information about Tulkka is here.

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 $L^q$ for Stokes operators with Hodge, Navier, and Robin type boundary conditions on uniform $C^{2,1}$ -domains $\Omega\subseteq\R^d$ . Our research complements recent results on the $L^q$ -theory of such operators and also sheds new light on the cases $q=1$ and $q=\infty$ .

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 $L^p$ 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 $\mathbb{R}^4$ 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
12:15-13:45		Lunch break
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
15:25-16:10		Coffee break
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 $H^1$ 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 $H^1$ regularity of the solution, $L^2$ -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 $L^p$ -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.

References:

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 $H^1$ -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 $L^2$ .

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 $L^p$ 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 $L^p$ spaces and relation to tent spaces
The theory of $L^p$ spaces for outer measures, or outer $L^p$ 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 $L^p$ 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 $\mathbb{R} \times (0,\infty)$ and in the upper half space $\mathbb{R}^2 \times (0,\infty)$ that are relevant in harmonic analysis. This allows us to clarify the relation between the outer $L^p$ spaces and the tent spaces introduced by Coifman, Meyer and Stein, and to get a glimpse at the use of the outer $L^p$ 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 $(X,\mu)$ be a measure space, where $X\subset \mathbb{R}^d$ and $\mu$ is doubling, and choose an orthonormal basis $\{\varphi_n\}_{n\in\mathbb{N}^d}$ in $L^2(X,\mu)$ . Hardy’s inequality on $H^p$ , $p \in (0,1]$ , associated with $\{\varphi_n\}$ has the following form
$\sum_{n\in\mathbb{N}^d}\frac{\lvert\langle f,\varphi_n\rangle\rvert^p}{(n_1+\dots+n_d+1)^E}\lesssim \lVert f\rVert_{H^p(X,\mu)}^p,\qquad f\in H^p(X,\mu),$
where $n=(n_1,\dots,n_d)$ , the symbol $\langle\cdot,\cdot\rangle$ stands for the inner product in $L^2(X,\mu)$ , and $H^p(X,\mu)$ denotes an appropriate Hardy space. Here $E$ 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 $X$ we investigate continuity properties of the semigroup that is induced in real interpolation spaces between $X$ and the domain $D(A)$ 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 $L=-\nabla\cdot\mu\nabla$ denote a second-order elliptic operator in divergence form and let $(e^{-tL})_{t\geq0}$ denote the corresponding strongly continuous heat semigroup on $L^2(\mathbb{R}^d)$ . If $E,F\subset\mathbb{R}^d$ denote measurable sets with $\text{dist}(E,F)>0$ and if $f\in L^2(\mathbb{R}^d)$ is supported in $E$ , then by the strong continuity of the semigroup, one finds that
$\begin{equation*}\lVert e^{-tL}f\rVert_{L^2(F)}\to\lVert f\rVert_{L^2(F)}=0\quad\text{as}\quad t\to0. \end{equation*}$
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
$\begin{equation*}\lVert e^{-tL}f\rVert_{L^2(F)}\lesssim e^{-\tfrac{c\text{dist}(E,F)^2}{t}}\lVert f\rVert_{L^2(E)}.\end{equation*}$
In this talk, we study off-diagonal behavior of the generalized Stokes semigroup $(e^{-tA})_{t\geq0}$ that is generated on $L^2_{\sigma}(\mathbb{R}^d)$ by the generalized Stokes operator with bounded measurable coefficients $\mu$ , formally given by
$\begin{equation*}Au:=-\text{div}(\mu\nabla u)+\nabla\phi,\quad \text{div}(u)=0\quad \text{in }\mathbb{R}^d.\end{equation*}$
In contrast to the elliptic operator $L$ , the operator $A$ exhibits a non-local behavior due to the presence of the pressure function $\phi$ . This non-locality affects the non-local behavior of the generalized Stokes semigroup $e^{-tA}$ and it is not clear how fast the support of a divergence free vector field $f$ that is supported in a set $E$ 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 $h^{1,q}_b(M)$ for $1<q <\infty$ and $b>0$ on more general measured metric spaces $M$ satisfying only some mild geometric assumptions. It will turn out, similar to the euclidean case, that the dual space of $h^{1,q}_b(M)$ is the space of functions with (local) bounded mean oscillation to the parameters $q'$ and $b$ , which we will denote by $bmo^{q'}_b(M)$ . We will then show, that both the local Hardy spaces $h^{1,q}_b(M)$ and the spaces $bmo^{q'}_b(M)$ are independent of the parameters $q$ and $b$ , and will therefore all be denoted by $h^1(M)$ and $bmo(M)$ respectively. We will then conclude, that for $2<p<\infty$ the function space $L^p(M)$ is a complex interpolation space between $L^2(M)$ and $bmo(M)$ . Therefore, we get by duality, that for $1<p<2$ the space $L^p(M)$ is an interpolation space between $h^1(M)$ and $L^2(M)$ . With that in mind, we will finally give some criteria under which linear Operators are bounded from $h^1(M)$ to $L^1(M)$ .

You can find an overview of older talks here.