Emergence of structures and advantages in cross-diffusion systems

交叉扩散系统的结构和优点的出现

• 批准号: 411007140
• 负责人: Professor Dr. Michael Winkler
• 金额: --
• 依托单位: National Natural Science Foundation of China
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/411007140?language=en
• 关键词: Emergence; structures; advantages; cross; diffusion

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.

拟议项目的主要目的包括在发展的方法，分析定性方面的系统的演化偏微分方程建模交叉扩散过程。尽管有关这类系统的各类数学文献大量增加，但迄今为止，只有在局部或整体存在理论的水平上才能实现全面的理解，与定性行为有关的问题，特别是有界解的问题，迄今为止还没有得到广泛的回答。从应用背景中，特别是在生物学学科，如空间生态学或病毒学，但也在社会科学，在其数学核心包括各种taxis类型或其他形式的交叉扩散，可能耦合到退化扩散或者还耦合到确定交叉扩散通量的相应量的演变中的基本上不稳定的机制。这些特定的设置已经以这样一种方式被选择，即它们的具体数学结构潜在地允许通过适当的情境适应技术的发明来分别深入理解，但是在随后的步骤中可以很好地想象到更一般的框架的概括。特别关注的问题是，各个迁移机制和交叉扩散相互作用在多大程度上可以被证明对整个系统或至少其部分是有益的。这将在适当的数学问题公式中进行，这些数学问题公式适当地量化了在相应的应用环境中明显相关的现象，例如，相对于静态对应物，相应地运动的亚群的优势，或空间结构的迁移驱动的支持，或者出租车诱导的持续定位。除了部分解决经典问题，例如确定吸引力驱动的爆破是否可能发生，因此特别旨在开发新的方法论方法，这些方法论方法有可能捕获更微妙的定性效果。通过涉及不同程度的数学复杂性的许多挑战，而且还通过将计划的活动与参与的中国团体的活动紧密联系起来，此外，预计该项目将为特别是年轻的研究人员提供相当大的专业培训和合作机会。因此，拓宽视野，进一步扩大中德合作和友谊将成为该项目的另一个重要目标。