Analysis of chemotactic cross-diffusion in complex frameworks

近年来，关于趋化过程基本模型的文献迅速增长，已经建立了关于此类偏微分方程系统的数学特征的相当多的知识，在适当的功能分析方法的可用性允许构建全球现有解决方案的情况下，可以获得最全面的结果。本项目的目标在于分析进化系统其中趋化交叉扩散的特征成分嵌入到更复杂的框架中的方程。后者可能会变得明显，在修改的数学结构方面的相互作用ofcross-diffusion与扩散和信号的演变，如建议在最近的文献中的精细建模方法，或在与进一步的组件和机制的相互作用;这里，特别关注在逻辑类型增殖条件影响下的趋化性，以及从流体动力学中耦合到（Navier-）Stokes方程的趋化系统。构成了理解趋化交叉扩散的不稳定潜力所必需的自然先决条件，作为该项目的一个中心和经常性主题，还将调查各种类型的出租机制在多大程度上可以强制爆炸解决方案的发生;尽管关于自发结构形成，这种爆破现象通常被认为是趋化性系统的最显著和广泛无所不在的数学特征，但迄今为止，它们的严格检测仅限于非常特定的星座。随着应用到更复杂的模型，从而更现实的情况下，该项目的一个目的是开发新的和强大的技术来分析趋化性系统中爆炸解决方案的发生和属性的有远见的目标。

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.