10:00-11:00 José Carrillo （University of Oxford, UK）
Title: Nonlocal Aggregation-Diffusion Equations: entropies, gradient flows, phase transitions, and applications
Abstract: This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics, and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential.
The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L² framework, nonlinear diffusion needs the stronger L^{∞} topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.
11:00-12:00 Elio Espejo （University of Nottingham Ningbo China）
Title: Optimal critical mass for the two-dimensional Keller-Segel model with rotational flux terms
Abstract: The aim of this talk is to show that several important systems of partial differential equations arising in mathematical biology, fluid dynamics, and electrokinetics can be approached within a single model, namely, a Keller-Segel-type system with rotational flux terms. In particular, we establish sharp conditions on the optimal critical mass for having global existence and finite time blow-up of solutions in two spatial dimensions. Our results imply that the rotated chemotactic response can delay or even avoid the blow-up. The key observation is that for any angle of rotation, the resulting PDE system preserves a dissipative energy structure.
This talk is based on my joint research with Professor Hao Wu (Fudan University).
16:00-17:00 Jie Jiang（Innovation Academy for Precision Measurement Science and Technology, CAS）
Title: On a Keller-Segel System of Chemotaxis with Density-suppressed Motility
Abstract: In this talk, we would like to report our recent work on a Keller-Segel system of chemotaxis, featuring density-suppressed motility. This model was originally proposed by Keller and Segel in their seminal work in 1971, which models the cellular movements due to a local sensing mechanism. An extended model was also developed in some recent works of Biophysics to study the process of pattern formation, involving density-suppressed motility, which stands for a repressive effect of the signal (and hence of the cell density) on cell motility.
From a mathematical point of view, the model features signal-dependent motility, which may vanish as the concentration becomes unbounded, leading to a possible degenerate problem. Conventional energy methods can only deal with some special cases and the existence of classical solutions with generic motility functions (i.e., positive, non-increasing and asymptotically vanishing functions) is a long-standing open problem. Recently, we develop a systematic approach to study the global existence as well as uniform boundedness of global solutions to this problem. The novelty of our method lies in an introduction of an auxiliary elliptic problem, which enables us to derive explicit point-wise upper bound estimates for the concentration via a delicate comparison argument. Furthermore, we can establish the uniform-in-time estimates by an Alikakos-Moser type iteration and an application of the theory for parabolic evolution operators. Our results indicate that this system permits global existence of classical solutions with generic motility functions in any dimension (thus no finite-time blowup), and the boundedness of solutions is closely related to dimensions and the decay rate of the motility function at infinity. In particular, a critical mass phenomenon as well as an infinite-time blowup was verified in the two-dimensional case.
The talk is based on my recent joint works with Kentaro Fujie (Tohoku University), Philippe Laurençot (University of Toulouse and CNRS), and Yanyan Zhang (ECNU).
17:00-18:00 Jaewook Ahn （Dongguk University, Korea）
Title: Asymptotics of PDEs arising from chemotaxis
Abstract: As the formation of singularity or instability in the chemotaxis PDEs can be seen as a prototype of a spontaneous biological pattern, it might be worthy to determine which conditions will cause the patterns to either persist or disappear. In this talk, we consider two types of logarithmic chemotaxis models featuring aggregation and pattern formation. As for the first model, a global weak solution is constructed using a Lyapunov structure. The constructed solution becomes smooth after some waiting time and stabilizes to a constant steady-state under further assumptions on the domain and the system parameters. As for the second model, we construct a global classical solution using a convenient energy method and find the condition on system parameters that make a constant steady-state globally stable or linearly unstable. Related works on the non-existence of non-constant steady states would be also discussed.
18:00-19:00 Michael Winkler（Paderborn University, Germany)
Title: Can primitive chemotaxis generate spatial structures?
Abstract: Parabolic models for the collective behavior in populations of single-cell species are considered. A predominant emphasis will be on cases in which individuals are particularly primitive in the sense that beyond a partially oriented movement toward increasing concentrations of a nutrient, further activity can essentially be neglected. Recent developments in the analysis of such nutrient taxis systems are to be described, with a special focus set on mathematical challenges related to the fundamental question how far models of this type are capable of adequately reflecting aspects of colorful dynamics known from experimental observations.
Time/Location: Friday, 10th September, at 10:00 – 19:00 in TB 405