Mini-workshop: Analysis, Numerics and Control

Date: Thursday, May 11, 2023:
Event: FAU DCN-AvH mini-workshop: Analysis, Numerics and Control
Organized by: FAU DCN-AvH, Chair for Dynamics, Control, Machine Learning and Numerics – Alexander von Humboldt Professorship at FAU, Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)

Title: Optimal control for suppression of singularity in chemotaxis
Speaker: Prof. Dr. Weiwei Hu
Affiliation: Visiting Scientist at FAU DCN-AvH from University of Georgia, Athens (UGA)
Abstract. In this talk, we discuss the problem of optimal control design for suppression of singularity via flow advection in chemotaxis modeled by the Patlak-Keller-Segel (PKS) equations. It is well-known that for the system without advection, singularity of the solution may develop at finite time. Specifically, if the initial condition is above certain critical threshold, the solution may blow up at finite time by concentrating positive mass at a single point. In this talk, we will first address the global regularity and stability of the PKS system in the presence of flow advection in a bounded domain, by using a semigroup approach. Then we focus on the design of an optimal flow field for suppressing such singularities. Rigorous theoretical framework and numerical experiments will be presented to demonstrate the ideas.

Title: Asymptotic behavior of solutions for some diffusion problems on metric graphs
Speaker: Prof. Dr. Liviu Ignat
Affiliation: Visiting Scientist at FAU DCN-AvH from Simion Stoilow Institute of Mathematics of the Romanian Academy
Abstract. In this talk we present some recent result about the long time behavior of the solutions for some diffusion processes on a metric graph. We study evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat equation (or even some nonlocal diffusion problems) is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. We prove that when time is large the solution behaves like a gaussian profile on the infinite edges. When the nonlinear convective part is present we obtain similar results but only on a star shaped tree.
Acknowledgment: this is a joint work with Cristian Cazacu (University of Bucharest), Ademir Pazoto (Federal University of Rio de Janeiro), Julio D. Rossi (University of Buenos Aires) and Angel San Antolin (University of Alicante).

Title: Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space
Speaker: Prof. Dr. Dragoș Manea
Affiliation: Visiting PhD student at FAU DCN-AvH from Simion Stoilow Mathematical Institute of the Romanian Academy,
Abstract. We study a non-local, non-linear convection-diffusion equation on the hyperbolic space \mathbb{H}^N, governed by two kernels, one for each of the diffusion and convection parts. One main novelty is the construction of the non-symmetric convection kernel defined on the tangent bundle and invariant under the geodesic flow. Next, we consider the relaxation of this model to a local problem, as the kernels get concentrated near the origin of each tangent space. Under some regularity and integrability conditions, we prove that the solution of the concentrated non-local problem converges to that of the local convection-diffusion equation.
We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain-Brezis-Mironescu.

Title: Gradient flow for the Kohn-Vogelius functional in shape optimization.
Speaker: Prof. Dr. Jan Sokolowski
Affiliation: Visiting Scientist at FAU DCN-AvH from IECL, Université de Lorraine, Nancy, France and Systems Research Institute of the Polish Academy of Sciences, Warsaw, Poland.
Abstract. The shape optimization problems can be solved by using the shape gradients
along with the shape Hessians, the phase field approach as well as the leveleset method.
In the review paper published in JGEA (2023), all possiblities of solution are listed for the solid and fluid mechanics applications of shape optimization.
The convergence proof of the solution strategy is only possible at the moment for the gradient flow method in the case of a model problem in two spatial dimensions. This result is presented in my talk, The new results are obtained in collaboration with Pavel I. Plotnikov.


On-site / Online

Room Übung 1. Department Mathematik.
FAU, Friedrich-Alexander-Universität Erlangen-Nürnberg.
Cauerstraße 11, 91058 Erlangen.

Zoom meeting link
Meeting ID: 614 4658 1599 | PIN code: 914397

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