Emergence of structures and advantages in cross-diffusion systems

• Applicant: Professor Dr. Michael Winkler
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 411007140

• Final Report Year: 2024

Final Report Abstract

This project aimed at developing methods capable of efficiently analyzing models for cross-diffusive processes with regard both to application-relevant properties and to theoretically motivated questions. Accordingly, the particular objects under investigation have been various classes of cross-diffusion systems proposed in contemporary modeling literature, and beyond addressing topics from existence and qualitative theories of immediate relevance in the respective application contexts, the project activities were partially also devoted to the derivation of results potentially suitable to contribute to a deeper understanding of mathematical properties of the problems under consideration. The main findings either characterize situations in which the respective models support structures and advantages, or identify conditions on the system constituents as sufficient for the prevalence of spatial homogeneity. This was achieved by analyzing the corresponding evolution equations firstly with regard to questions from local and global existence theories, and secondly with respect to aspects of qualitative solution behavior, either in the large time limit or near times and places of possible explosions. Particular contexts in which some progress could thereby be achieved range from simple two-component Keller-Segel type models, over parabolic systems involving various types of cross-diffusion mechanisms with more intricate and mathematically delicate couplings, to yet more complex migration models containing further components and interaction mechanisms.

Publications

• Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?. International Mathematics Research Notices, 2021(11), 8106-8152.
  Winkler, Michael
  
• Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow. Transactions of the American Mathematical Society, 374(1), 219-268.
  Winkler, Michael
  
• A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation. Journal of Functional Analysis, 281(4), 109069.
  Tao, Youshan & Winkler, Michael
  
• Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary. Communications in Partial Differential Equations, 46(6), 1058-1091.
  Wang, Yulan; Winkler, Michael & Xiang, Zhaoyin
  
• Reaction-Driven Relaxation in Three-Dimensional Keller–Segel–Navier–Stokes Interaction. Communications in Mathematical Physics, 389(1), 439-489.
  Winkler, Michael
  
• A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system. Journal de Mathématiques Pures et Appliquées, 162, 124-151.
  Fuhrmann, Jan; Lankeit, Johannes & Winkler, Michael
  
• A family of mass‐critical Keller–Segel systems. Proceedings of the London Mathematical Society, 124(2), 133-181.
  Winkler, Michael
  
• Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?. Journal of the European Mathematical Society, 25(4), 1423-1456.
  Winkler, Michael
  
• Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System. SIAM Journal on Mathematical Analysis, 54(4), 4806-4864.
  Tao, Youshan & Winkler, Michael
  
• Taxis-driven persistent localization in a degenerate Keller-Segel system. Communications in Partial Differential Equations, 47(12), 2341-2362.
  Stevens, Angela & Winkler, Michael