Oberseminar AG Mathematische Modellierung

Montag, den 25.11.2024 um 17.00 Uhr in B7 - 210 - Seminarraum A

Vortrag von Pierre Marie Ngougoue Ngougoue (Clausthal) zum Thema „Random elliptic quasi-variational inequalities and risk-averse optimal control“.

Titel:  

We investigate a risk-averse optimal control problem governed by an elliptic quasi-variational inequality (QVI) subject to random inputs. By extending existing methods in stochastic mathematical programs with equilibrium constraints (SMPECs), we introduce a penalty-based regularisation of the obstacle problem yielding a well-defined regularised control problem. This regularisation maintains consistency with the primal control problem and provides a robust framework for further analysis. Building on this, we derive the stationarity conditions for the regularised, smoothed problem and, through a limiting analysis with respect to the regularisation parameter, establish C-stationarity conditions for the primal control problem. The introduction of uncertainty…

Oberseminar AG Mathematische Modellierung

Montag, den 04.11.2024 um 17.00 Uhr in B7 - 210 - Seminarraum A

Vortrag von Anatole Gaudin (Clausthal) zum Thema „An introduction to Besov spaces through the (damped) heat equation“.

Abstract: The main goal of this talk is to introduce Besov spaces, which are a natural generalization of Sobolev spaces, allowing more flexibility in the study of PDEs. They provide a finer way to measure integrability and regularity than Sobolev spaces, and they also offer more flexibility for establishing effective non-linear estimates.

To motivate our first definition and use of Besov spaces $B^{s}_{p,q}(\mathbb{R}^n)$, we will investigate the behavior of the linear (damped) heat equation with respect to the initial data, first in the spaces $L^p_{t,x}(\mathbb{R}_+\times\mathbb{R}^n)$ and then in $L^q_{t}(\mathbb{R}_+, L^p_{x}(\mathbb{R}^n))$, with $1\leqslant p,q\leqslant + \infty$. We will first treat the case $p=2$, introducing fractional Sobolev spaces $\H^{s}(\mathbb{R}^n)$.

Equivalent…

Oberseminar AG Mathematische Modellierung

Montag, den 28.10.2024 um 17.00 Uhr in B7 - 210 - Seminarraum A

Vortrag von Dr. Pei Su (Paris-Saclay) zum Thema „Conditional regularity for an elastic shell interacting with the Navier-Stokes equations“.

Abstract: We are interested in the interaction of a viscous incompressible fluid with an elastic structure, where the structure is located on a part of the fluid boundary. It reacts to the surface forces induced by the fluid and deforms the reference domain to the moving domain. The fluid equations are coupled with the structure via the kinematic condition and the action-reaction principle on the interface.

We study the 2D visco-elastic shell interacts with 3D Navier-Stokes equations. Especially in a general reference geometry (the shell deforms along the normal direction of the flexible boundary), we prove a counterpart of the classical Ladyzhenskaya-Prodi-Serrin condition yielding conditional regularity and uniqueness of a solution. This requires additionally the deformation of…

Oberseminar AG Mathematische Modellierung

Montag, den 21.10.2024 um 17.00 Uhr in B7 - 210 - Seminarraum A

Vortrag von Prof. Dr. Dominic Breit zu dem Thema: „The Stokes problem with Navier boundary conditions in irregular domains“.

Abstract: 

We consider the steady Stokes equations supplemented with Navier boundary conditions including a non-negative friction coefficient. We prove maximal regularity estimates (including the prominent spaces $W^{1,p}$ and $W^{2,p}$ for $p\in(1,\infty)$ for the velocity field) in bounded domains of minimal regularity. Interestingly, exactly one derivative more is required for the local boundary charts compared to the case of no-slip boundary conditions. We demonstrate the sharpness of our results by a propos examples.

This is joint work with Sebastian Schwarzacher.