Concentration Phenomena in Diffusion and Cross-Diffusion Systems

题目：扩散和交叉扩散系统中的浓度现象spi：倪伟明（明尼苏达大学-双城分校）(I)技术描述：倪教授计划继续他的研究，以数学方式理解各种扩散相关机制的影响。利用经典分析中的变分方法和技术，尼教授和他的合作者已经建立了形态发生中活化剂-抑制剂系统的尖峰层稳态，该系统是由Gierer和meinhardt根据图灵1952年著名的扩散驱动不稳定性理论提出的。该方法利用了两种化学物质的扩散速率之间的差距，并在一个空间维度上获得了稳定性结果。最近，在多维子集上的集中现象方面取得了进展。然而，完整的动态仍远未被完全理解。已经注意到，仅靠扩散速率之间的差距不足以形成模式。因此，“交叉扩散”的概念，由理论生物学家于1979年引入，用于模拟种群动力学中的隔离现象，值得系统研究。交叉扩散系统近年来也被用于模拟奇异现象，包括细菌菌落的树突生长，它在最高阶项上既是非线性的，又是强耦合的，因此在数学上非常具有挑战性。倪教授和他的合作者建议研究交叉扩散的影响，首先获得交叉扩散的必要和充分条件，以帮助创造图案，然后研究它们的定性行为和稳定性。（二）非技术描述：倪教授计划继续他的研究，以严谨的数学方式理解各种扩散相关机制的现象和影响，并希望以此对提高我们在应用科学中模拟更复杂和/或现实现象的能力，以及创造新的和有意义的数学产生一些影响。在本提案中，倪教授打算从模式形成的角度研究扩散和/或交叉扩散系统中的各种“集中现象”。这些，特别是包括化学反应中的图灵模式（例如CIMA反应），水螅形态发生中模拟再生现象的Gierer-Meinhardt的激活剂-抑制剂系统，模拟细菌菌落树突状生长的非线性扩散系统，以及考虑到种间种群压力的otka- volterra竞争系统。”

Proposal DMS-0400452Title: Concentration phenomena in diffusion and cross-diffusion systemsPI: Wei-Ming Ni (University of Minnesota- Twin Cities)(I) Technical Description:Professor Ni plans to continue his research in understandingmathematically the effects of various diffusion-related mechanisms. Usingvariational approach and techniques in classical analysis, Professor Niand his collaborators have established the spike-layer steady states foran activator-inhibitor system in morphogenesis proposed by Gierer andMeinhardt based on the celebrated idea - diffusion-driven instability - ofTuring in 1952. The method exploits the gap between the diffusion ratesfor the two chemical substances, and stability results in one spacedimension have also been obtained. Very recently, progress has been madefor concentration phenomena on multi-dimensional subsets. However, thecomplete dynamics is still far from being fully understood. It has beennoted that the gap between the diffusion rates alone is insufficient tocreate patterns. Thus, the notion of "cross-diffusion," introduced bytheoretical biologists in 1979 in modeling segregation phenomena inpopulation dynamics, deserves systematic studies. Cross-diffusion systems,which have also been used in recent years to model singular phenomenaincluding dendritic growth of bacteria colonies, are both nonlinear andstrongly-coupled in highest order terms, thus are mathematically verychallenging. Professor Ni and his collaborators propose to study theeffects of cross-diffusion by first obtain necessary and sufficientconditions for cross-diffusion to help create patterns, and then toinvestigate their qualitative behavior as well as their stabilityproperties.(II) Non-technical Descriptions:Professor Ni plans to continue his research in understanding, in amathematically rigorous manner, the phenomena and effects of variousdiffusion-related mechanisms and hopefully thereby to have some impact inboth improving our ability in modeling more complicated and/or realisticphenomena in applied sciences, as well as in creating new and significantmathematics. In this proposal, from the viewpoint of pattern formation,Professor Ni intends to investigate the various "concentration phenomena"in diffusion and/or cross-diffusion systems. These, in particular, includeTuring patterns in chemical reactions (e.g. the CIMA reaction),Gierer-Meinhardt's activator-inhibitor systems in modeling theregeneration phenomena of hydra in morphogenesis, a nonlinear diffusionsystem modeling dendritic growth of bacteria colonies, and theLotka-Volterra competition systems with inter-specific populationpressures taken into considerations."