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Events
Dresden-Leipzig AG seminar day

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig)

Inselstr. 22
04103 Leipzig
Germany

E1 05 (Leibniz-Saal)

conference

18.10.24

Dresden-Leipzig AG seminar day

More information about this event will follow soon.

Speakers

Guy Fabrice Foghem Gounoue

TU Dresden

Kai Richter

TU Dresden

Anton Ullrich

MPI MIS

Program

11:00 - 12:00	Guy Fabrice Foghem Gounoue (TU Dresden) Gradient flow solutions for porous medium equations with pressure driven by nonlocal symmetric Lévy operators We examine a nonlocal porous medium equation, $\partial_t u + \operatorname{div}(u \nabla L^{-1} u) = 0$, where the pressure is governed by a general symmetric Lévy operator, $L$, which is widely recognized as the generator of a symmetric Lévy stochastic process. Namely, we consider a nonlocal operator of the form \begin{equation} Lu(x)=\operatorname{p.v.}\int_{\mathbb{R}^d} (u(x)-u(y))\nu(x-y)\mathrm{d} y (x\in\mathbb{R}^d), \end{equation} where $\min(1,\|h\|^2)\nu\in L^1(\mathbb{R}^d)$ and $\nu(h)=\nu(-h)$. A nonlocal symmetric Lévy operator generalizes the classical fractional Laplace operator $(-\Delta)^s $, with $s \in (0,1) $. We construct weak solutions in the context of the corresponding nonlocal Sobolev space using the Jordan-Kinderlehrer-Otto (JKO) minimizing movement scheme. The absence of interpolation and various tools from classical fractional Sobolev spaces renders our approach more challenging. Furthermore, we shall investigate the nonlocal-to-local convergence of the problem.
12:00 - 12:15	Discussion/Talk
12:15 - 13:15	Lunch
13:15 - 14:15	Anton Ullrich (MPI MIS) tba
14:15 - 14:30	Discussion/Talk
14:30 - 15:00	Coffee Break
15:00 - 16:00	Kai Richter (TU Dresden) Dimension reduction for elastoplastic rods in the bending regime In our work we rigorously derive a limiting model for thin rods, starting from a full 3D model for finite plasticity. We are interested in a scaling of the elastic and plastic energy contributions like h^(-4), where h denotes the thickness of the rod. In the limit this results in a 1D bending theory. For the derivation we lean on the framework of evolutionary Gamma-convergence for rate-independent systems, introduced by Mielke, Roubíček and Stefanelli in 2008. The main difficulty here is to establish a mutual recovery sequence for the stored energy and dissipation. Strategies have been developed by various authors in order to construct such a sequence, e.g. for linearization or in the von Kármán regime. However, these rely on considering infinitesimal deformations in the limit, which we cannot expect in the bending regime. Our approach relies on a construction based on a multiplicative decomposition of the rotation fields obtained via the rigidity estimate from Friesecke, James and Müller. In order to achieve enough regularity, we consider strain gradient terms in the energy, which act on the two parts of the polar decomposition individually. These terms vanish in the limit. This is joined work with Stefan Neukamm.
16:00 - 16:15	Discussion/Talk

Dresden-Leipzig AG seminar day

Speakers

Guy Fabrice Foghem Gounoue

Kai Richter

Anton Ullrich

Program

Participants

Guy Fabrice Foghem Gounoue

Kai Richter

Elena Salguero

Anton Ullrich

Organizers

Elena Salguero

Administrative Contact

Katja Heid