Concentration Phenomena in Diffusion and Cross-Diffusion Systems

提案DMS-0400452题目：扩散和交叉扩散系统中的浓度现象PI：Wei-Ming Ni（明尼苏达大学双城分校）（I）技术说明：Ni教授计划继续他的研究，从数学上理解各种扩散相关机制的影响。在1952年Turing提出的扩散驱动不稳定性的基础上，倪教授和他的合作者利用经典分析中的变分方法和技巧，建立了Gierer和Meinhardt提出的形态发生中激活-抑制系统的尖峰层定态。该方法利用了两种化学物质扩散速率之间的差距，并且在一维空间中也得到了稳定的结果。最近，多维子集上的集中现象研究取得了进展。然而，完整的动力学仍然远远没有被完全理解。已经注意到，仅扩散速率之间的差距不足以形成图案。因此，1979年理论生物学家在模拟种群动态中的分离现象时引入的“交叉扩散”概念值得系统研究。交叉扩散系统是一类非线性的高阶强耦合系统，近年来也被用来模拟细菌菌落的树枝状生长等奇异现象，因此在数学上具有很大的挑战性.倪教授和他的合作者们建议研究交叉扩散的影响，首先获得交叉扩散的必要条件，以帮助创建模式，然后研究它们的定性行为以及它们的稳定性。(II)非技术性描述：倪教授计划继续他的研究，以数学严谨的方式理解各种扩散相关机制的现象和影响，并希望由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重要的数学产生一定的影响。在这个提议中，倪教授打算从图案形成的角度来研究扩散和/或交叉扩散系统中的各种“集中现象“。这些，特别是，包括图灵模式的化学反应（如CIMA反应），Gierer-Meinhardt的激活剂-抑制剂系统在模拟再生现象的水螅在形态发生，一个非线性扩散系统模拟树突状生长的细菌菌落，和Lotka-Volterra竞争系统与种间populationpressure考虑在内。"

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."