Analysis of chemotactic cross-diffusion in complex frameworks

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In the recent years, a rapidly growing literature on basic models for chemotaxis processes has established a considerable knowledge on mathematical features of such systems of partial differential equations, where the most comprehensive results could be achieved in situations when the availabilityof appropriate functional analytical approaches allow for the construction of globally existing solutions.The goal of the present project consists in analyzing systems of evolution equations in which the characteristic component of chemotactic cross-diffusion is embedded into more complex frameworks. The latter may either become manifest in modified mathematical structures with regard to the interplay ofcross-diffusion with diffusion and signal evolution, as suggested by refined modeling approaches in the recent literature, or in interaction with further components and mechanisms; here, a particular focus will be on chemotaxis under the influence of logistic-type proliferative terms, and on chemotaxis systems coupled to the (Navier-)Stokes equations from fluid dynamics.Constituting a natural prerequisite essential for the understanding of the destabilizing potential of chemotactic cross-diffusion, as a central and recurrent theme of the project it will moreover be investigated how far various types of taxis mechanisms may enforce the occurrence of exploding solutions; although with regard to spontaneous structure formation such blow-up phenomena are commonly regarded as the most striking and widely omnipresent mathematical feature of chemotaxis systems, their rigorous detection is limited to very particular constellations up to now. With the visionary goal of application to more complex models and thus to more realistic situations, one intention of the project is to develop new and robust techniques for analyzing the occurrence and properties of exploding solutions in chemotaxis systems.