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Forschungsseminar (Dauerveranstaltung)

Vorträge im Sommersemester 2023

Die Vorträge finden im Seminarraum 2.066 im Mathematikgebäude 20.30 statt.

25.04.2023, 14:00 Uhr Maximilian Ruff (Karlsruhe)

Lie splitting for semilinear wave equations at H^1 regularity
We consider the semilinear wave equation in \mathbb{R}^3 with energy-(sub)critical power nonlinearity, and analyze a frequency-filtered Lie splitting scheme for the semidiscretization in time. For initial data in the energy space H^1 \times L^2, we prove first-order convergence in L^2 \times H^{-1}. The error analysis relies on time-discrete Strichartz estimates for the wave propagator.

02.05.2023, 14:00 Uhr Luca Haardt (Karlsruhe)

On well-posedness of parabolic equations of Navier-Stokes type with BMO^{-1} data
In this talk I will present the topic of my master thesis which is based on a paper by Auscher and Frey. They reproved the well-posedness result of Koch and Tataru of the incompressible Navier-Stokes equations with initial data in BMO^{-1} by tools coming from harmonic analysis. Moreover, they adapted their new proof to show even well-posedness of parabolic equations with similar quadratic nonlinearities but rougher second-order elliptic operators L = -\operatorname{div}(A\nabla \cdot) with bounded measurable coefficients for initial data in the operator-adapted space BMO^{-1}_L.

09.05.2023, 15:45 Uhr Robert Schippa (Karlsruhe)

Oscillatory integral operators with homogeneous phase functions
We consider oscillatory integral operators with 1-homogeneous phase functions satisfying a convexity condition. These generalize the cone extension operator. We show L^p-L^p-estimates via polynomial partitioning and decoupling estimates. The L^p-estimates are sharp up to endpoints, which follows from examples of operators exhibiting Kakeya compression. We use the estimates to prove new local smoothing estimates for wave equations on Riemannian manifolds. The talk is based on arXiv:2109.14040 (accepted to Journal d'Analyse Mathématique).

16.05.2023, 14:00 Uhr Fabian Gabel (Hamburg)

The Finite Section Method and Periodic Schrödinger Operators
We study discrete Schrödinger operators $H$ on $\ell^p(\mathbb{Z})$ with periodic potentials as they are typically used to approximate aperiodic Schrödinger operators like the Fibonacci Hamiltonian. We prove an efficient test for applicability of the finite section method, a procedure that approximates $H$ by growing finite square submatrices $H_n$. The study of the applicability of the finite section method also gives further insights on the location of Dirichlet eigenvalues of half-line Schrödinger operators on $\ell^p(\mathbb{Z}_+)$. This talk is based on the findings in arXiv:2110.09339 (to appear in Operator Theory: Advances and Applications) and the analysis code doi:10.15480/336.3828.

23.05.2023, 14:00 Uhr Peer Kunstmann (Karlsruhe)

Minimal periods for semilinear parabolic equations
We show that, if $-A$ generates a bounded holomorphic semigroup in a Banach space X, $ \alpha\in[0,1) $, and $ f: D(A) \to X $ satisfies $\| f(x) - f(y) \| \le L \| A^\alpha ( x - y ) \|$, then a non-constant $T$-periodic solution of the equation $ u'(t) + A u(t) = f ( u(t) ) $ satisfies $ L T^{1-\alpha} \ge K_\alpha $, where $ K_\alpha > 0 $ is a constant depending on $\alpha$ and the semigroup. This extends results by Robinson and Vidal-Lopez, which have been shown for self-adjoint operators $ A \ge 0 $ in Hilbert space. For the latter case, we obtain the optimal constant $ K_\alpha $, which only depends on $\alpha$, and we also include the case  $\alpha = 1$. This is joint work with Gerd Herzog (Karlsruhe).

20.06.2023, 14:00 Uhr Jonas Sauer (Jena)


18.07.2023, 14:00 Uhr


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