|C: Models and Approximations|
C1: Evolution and asymptotics
C2: Multi-scale phenomena and macroscopic structures
|Current Publications||• Ceci, Stefano; Seis, Christian On the dynamics of vortices in viscous 2D flows. Mathematische Annalen Vol. online first, 2023 online|
• Choi, Beomjun; McCann, Robert J.; Seis, Christian Asymptotics near extinction for nonlinear fast diffusion on a bounded domain. Archive for Rational Mechanics and Analysis Vol. 247, 2023 online
• Beck, Lisa; Cinti; Eleonora; Seis, Christian Optimal regularity of isoperimetric sets with Hölder densities. , 2023 online
• Schlichting, André; Seis, Christian The Scharfetter-Gummel scheme for aggregation-diffusion equations. IMA Journal of Numerical Analysis Vol. 42 (3), 2022 online
• Seis C Bounds on the rate of enhanced dissipation. Communications in Mathematical Physics Vol. Online first, 2022 online
• Nobili, Camilla; Seis, Christian Renormalization and energy conservation for axisymmetry fluid flows. Mathematische Annalen Vol. 382, 2022 online
• Navarro-Fernández V, Schlichting A, Seis C Optimal stability estimates and a new uniqueness result for advection-diffusion equations. Pure and Applied Analysis Vol. 4 (3), 2022 online
• Meyer, David; Seis, Christian Propagation of regularity for transport equations. A Littlewood-Paley approach. arXiv Preprit:2203.10860 Vol. 2022, 2022 online
• Ceci S, Seis C On the dynamics of point vortices for the 2D Euler equation with Lp vorticity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Vol. 380 (2226), 2022 online
|Current Projects||• Transport Equations, mixing and fluid dynamics Advection-diffusion equations are of fundamental importance in many areas of science. They describe systems, in which a quantity is simultaneously diffused and advected by a velocity field. In many applications these velocity fields are highly irregular. In this project, several quantitative aspects shall be investigated. One is related to mixing properties in fluids caused by shear flows. The interplay between the transport by the shear flow and the regularizing diffusion leads after a certain time, to the emergence of a dominant length scales which persist during the subsequent evolution and determine mixing rates. A rigorous understanding of these phenomena is desired. In addition, stability estimates for advection-diffusion equations will be derived. These shall give a deep insight into how solutions depend on coefficients and data. The new results shall subsequently be applied to estimate the error generated by numerical finite volume schemes approximating the model equations. online|
• EXC 2044 - C1: Evolution and asymptotics In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.
Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues.
Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees.For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities. online
• EXC 2044 - C2: Multi-scale phenomena and macroscopic structures In multi-scale problems, geometry and dynamics on the micro-scale influence structures on coarser scales. In this research unit we will investigate and analyse such structural interdependence based on topological, geometrical or dynamical properties of the underlying processes.
We are interested in transport-dominated processes, such as in the problem of how efficient a fluid can be stirred to enhance mixing, and in the related analytical questions. A major concern will be the role of molecular diffusion and its interplay with the stirring process. High Péclet number flow in porous media with reaction at the surface of the porous material will be studied. Here, the flowinduces pore-scale fluctuations that lead to macroscopic enhanced diffusion and reaction kinetics. We also aim at understanding advection-dominated homogenisation problems in random regimes.
We aim at classifying micro-scale geometry or topology with respect to the macroscopic behaviour of processes considered therein. Examples are meta material modelling and the analysis of processes in biological material. Motivated by network formation and fracture mechanics in random media, we will analyse the effective behaviour of curve and free-discontinuity energies with stochastic inhomogeneity. Furthermore, we are interested in adaptive algorithms that can balance the various design parameters arising in multi-scale methods. The analysis of such algorithms will be the key towards an optimal distribution of computational resources for multi-scale problems.
Finally, we will study multi-scale energy landscapes and analyse asymptotic behaviour of hierarchical patterns occurring in variational models for transportation networks and related optimal transport problems. In particular, we will treat questions of self-similarity, cost distribution, and locality of the fine-scale pattern. We will establish new multilevel stochastic approximation algorithms with the aim of numerical optimisation in high dimensions. online
|Phone||+49 251 83-35142|
|FAX||+49 251 83-32729|
|Secretary||Sekretariat Claudia Giesbert|
Frau Claudia Giesbert
Telefon +49 251 83-33792
Fax +49 251 83-32729
|Address||Prof. Dr. Christian Seis |
Angewandte Mathematik Münster: Institut für Analysis und Numerik
Fachbereich Mathematik und Informatik der Universität Münster
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