Emergence of structures and advantages in cross-diffusion systems

拟议项目的主要目的在于开发分析交叉扩散过程的演化偏微分方程组中的定性方面的方法。尽管与这类系统相关的数学文献有了相当大的增长，但到目前为止，人们只能在局部或整体存在理论的水平上获得全面的理解，有关定性行为的问题，特别是有界解的问题，到目前为止还没有得到回答。需要研究的具体问题来自应用背景，特别是在空间生态学或病毒学等生物学学科中，但也在社会科学中，其数学核心包括各种趋势型或其他形式的交叉扩散，可能耦合到退化扩散，或者也耦合到决定交叉扩散通量的各个量的演化中的实质上的不稳定机制。这些具体设置的选择方式是，它们的具体数学结构可能允许通过发明适当的适应情况的技术来分别深入理解，但在随后的步骤中很容易想到对更一般框架的概括。除了与规律性和有界性主题有关的自然先决条件之外，特别关注的问题是，可以在多大程度上证明各自的迁移机制和交叉扩散相互作用对整个系统或至少对其中的一部分是有益的。这将在适当量化在各自应用环境中明显相关的现象的适当数学问题公式中进行，例如，相应的活动亚群相对于静态对应群体的优势，或者迁移驱动的空间结构的支持，或者出租车引起的持久本地化。除了部分解决经典问题，例如确定是否会发生吸引力驱动的爆炸，因此特别旨在开发新的方法方法，有可能捕捉到更微妙的定性影响。通过涉及不同程度的数学复杂性的众多挑战，而且通过将计划的活动与参与的中国小组的活动紧密联系起来，该项目还有望为特别是年轻的研究人员提供相当大的专业培训和合作的可能性。因此，拓宽视野，进一步扩大中德合作与友谊，将成为拟议项目的另一个重要目标。

The main purpose of the proposed project consists in developing methods of analyzing qualitative aspects in systems of evolutionary partial differential equations modeling cross-diffusion processes. Despite a considerable increase of the mathematical literature concerned with various classes of such systems, a comprehensive understanding could so far be achieved only at levels of local or global existence theory, with questions related to qualitative behavior especially of bounded solutions widely left unanswered up to now.The particular problems to be studied, taken from application contexts especially in disciplines of biology such as spatial ecology or virology, but also in social sciences, at their mathematical core include various taxis-type or other forms of cross-diffusion, possibly coupled to degenerate diffusion or also to substantially destabilizing mechanisms in the evolution of the respective quantity determining the cross-diffusive flux. These specific settings have been chosen in such a way that their concrete mathematical structure potentially allows for a respectively deep understanding through the invention of appropriately situation-adapted techniques, but that generalizations to more general frameworks are well conceivable in subsequent steps.Beyond natural prerequisites related to topics of regularity and boundedness, a particular focus will be on the question how far the respective migration mechanisms and cross-diffusive interactions can be proved beneficial for the overall system or at least for parts thereof. This is to be pursued within suitable mathematical problem formulations appropriately quantifying phenomena of apparent relevance in the respective application context, such as dominance of accordingly motile subpopulations over static counterparts, or migration-driven support of spatial structures, or also taxis-induced persistent localization, for instance. Apart from partially addressing classical issues, e.g. of determining whether attraction-driven blow-up may occur, it is thereby particularly intended to develop new methodological approaches with potential to capture more subtle qualitative effects.By involving numerous challenges at various degrees of mathematical intricacy, but also by closely connecting the planned activities with those of the participating Chinese group, this project can moreover be expected to provide especially young researchers with considerable possibilities of professional training and collaboration. Widening perspectives and further extending the Sino-German cooperation and friendship will thus form a further vital objective of the proposed project.