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Research Seminar

Research Seminar (Continuing Class)

Talks in winter 2024/25

The talks are taking place in seminar room 2.066.

26.11.2024, 14:00 Uhr Jürgen Saal (Düsseldorf)

Existence of a Global Attractor for Living Fluids
Self-sustained turbulent motion in microbiological suspensions present an intriguing example of collective dynamical behavior among the simplest forms of life and is important for fluid mixing and molecular transport on the microscale. This type so-called active or living fluids display turbulent behavior at low Reynolds regimes, a phenomenon that cannot be captured by classical fluid models. In a paper of Wensink et. al. a generalized version of the Navier-Stokes equations is proposed to describe this so-called 'active turbulence'. The purpose of the talk is to analyze the active turbulence from an analytical point of view. We will discuss (in-) stability of relevant equilibria and prove the existence of a global attractor.

10.12.2024, 14:00 Uhr David Ploß (Karlsruhe)

Maxwell’s Equation with Partially Absorbing Boundary Condition
This talk investigates well-posedness of Maxwell's equations on $\Omega=(0,T)\times \mathbb{R}^3_+$ with a special type of absorbing boundary conditions given by
\[E \times \nu = g-\beta(E \times \nu)(\nu \times (H \times \nu)). \]
A wide range of variants of absorbing boundary conditions where the damping $\beta$ is uniformly positive definite on the entire domain have been studied. In this talk, however, we investigate the case where $\beta$ is allowed to vanish on some part of the boundary, leaving the highly irregular condition $E \times \nu=g$ there.
Also $\beta$ needs to be $W^{1,\infty}$, so we have to deal with a smooth transition between the damped and the undamped region giving rise to weighted $L^2$-spaces, which are ill-compatible with the existing regularization approaches in the literature. This makes it especially challenging to obtain existence and sufficient regularity properties of the appearing boundary traces, which a-priori only have $H^{-1/2}$-regularity in space and time.
As $\big(H^{1/2}((0,T)\times \mathbb{R}^2)\big)' \neq H^{-1/2}((0,T)\times \mathbb{R}^2)$, we carefully need to establish estimates in the correct dual spaces.
By splitting the equation into several sub-problems, we finally obtain well-posedness of the linearized problem, which lies the foundation to further non-linear studies.

17.12.2024, 14:00 Uhr Christopher Bresch (Karlsruhe)

Local wellposedness on \mathbb R^3 of Maxwell systems with retarded material laws
Maxwell equations in matter have to be complemented with material laws describing the response of the material to electromagnetic fields. In this talk we consider a model commonly used in nonlinear optics, which includes retardation effects. Local wellposedness of this system in the space H^s with s>3/2 has been proven by A. Babin and A. Figotin (2003). To study local wellposedness for mild solutions in H^s with 1<s\le 3/2, we use a Strichartz estimate due to R. Schippa (2024) as well as a Helmholtz decomposition. The talk is based on joint work with Roland Schnaubelt.

07.01.2025, 14:00 Uhr Luca Haardt (Karlsruhe)

Kato's square root property for the generalized Stokes operator
In this talk we establish the Kato square root property for the generalized Stokes operator on $\mathbb{R}^d$ with rough coefficients. More precisely, we identify the domain of the square root of $Au := - \operatorname{div}(\mu \nabla u) + \nabla \phi$, $\operatorname{div}(u) = 0$, with the space of divergence-free $H^1$-vector fields and further prove the estimate $\| A^{1/2} u \|_{L^2} \simeq \| \nabla u \|_{L^2}$. As an application we show that $A^{1/2}$ depends holomorphically on the coefficients $\mu$. Except for boundedness and measurability as well as an ellipticity condition on $\mu$, there are no requirements on the coefficients.

14.01.2025, 14:00 Uhr Peer Kunstmann (Karlsruhe)

$L^q$-theory for Stokes equations with dynamic boundary conditions
In fluid mechanics of molten polymers, certain models involve dynamic boundary conditions of Navier type. Mathematical research activities on these problems are quite recent.
In this talk we establish maximal $L^p$-regularity for the linear Stokes system with dynamic boundary conditions in an $L^q$-setting for $1<p,q<\infty$ on bounded $C^{2,1}$-domains $\Omega\subseteq\mathbb{R}^d$. We also prove local wellposedness for a corresponding Navier-Stokes system.
This is joint work with H. Kozono (Waseda University and Tohoku University) and S. Shimizu (Kyoto University).

28.01.2025, 14:00 Uhr Himani Sharma (Karlsruhe)

Bochner-Riesz means for the Laguerre expansion
In this talk we start by discussing about the Bochner-Riesz means and spectral multiplier results corresponding to the harmonic oscillator $L_H=-\frac{d^2}{dx^2}+x^2$ and Schrödinger operator with anharmonic potential, ${\mathcal L}=-\frac{d^2}{dx^2}+|x|$. We show that the Bochner-Riesz profile of the operator ${\mathcal L}$ completely coincides with such profile of the harmonic oscillator. Then we extend our discussion to include the Laguerre expansion of order $\alpha$ corresponding to the operator ${H_\alpha}=-\frac{d^2}{dx^2}-(2\alpha+1)\frac{d}{dx}+x^2$, which can be interpreted as a radial part of multidimensional harmonic oscillators.

11.02.2025, 14:00 Uhr George Weiss (Tel Aviv)

The local representation of nonlinear infinite-dimensional well-posed systems and impedance passive nonlinear systems with monotone feedback
We investigate the local (in time) description of incrementally scattering passive nonlinear systems. We show that these systems can be defined by a differential inclusion and a function that gives the current output in term of the current state and the current input. Our approach uses the theory of maximal monotone operators and Lax-Phillips type nonlinear semigroups. We prove the well-posedness of a feedback system obtained by closing a feedback loop around an incrementally impedance passive, possibly nonlinear system. The static feedback operator is assumed to be maximal monotone. This result may lead to a local representation of incrementally impedance passive systems (this is work in progress).

18.02.2025 TULKKA in Konstanz (Poster, Abstracts, talks in room K 7)
11:45-12:30 Simon Bau (Konstanz) Evolution equations with dynamical boundary conditions in Banach scales
12:30-13:45 Lunch Break
13:45-14:30 Manuel Schlierf (Ulm) Gradient flow approaches to the Canham-Helfrich model
14:45-15:30 Siliang Weng (Karlsruhe) Phase space methods for magnetic evolution equations
15:30-16:15 Coffee Break
16:15-17:00 Yuxi Hu (Beijing) The initial boundary value problem for relaxed compressible Navier-Stokes equations
More information on Tulkka is here.

You can find previous talks in the archive of the research seminar.