Mathematical Sciences: Diffusion, Cross-Diffusion and Spike Layers

9705639 Ni教授计划继续他的研究，从数学上研究各种与扩散有关的机制。对这些现象和扩散效应的透彻理解将提高我们在应用科学中模拟更复杂或更现实情况的能力。在过去的十年里，似乎已经取得了实质性的进展，并且已经为更令人兴奋的数学突破做好了准备，特别是在与扩散相关的模式形成方面。在这个项目中，Ni教授将研究以下三个密切相关的方面：(I)半线性扩散方程系统（II）自扩散和交叉扩散方程系统（III）扩散/交叉扩散系统中的峰层（点凝聚解）。本项目的目的不仅是进一步分别发展这些方向，而且是综合所获得的结果，并建立一个理论，以提供一个系统的观点来理解微分方程中的“集中”现象。1979年，京都大学的几位日本科学家试图理解种群动态中的隔离现象，将“自扩散”和“交叉扩散”的概念纳入了Lotka-Volterra经典的两个竞争物种动态模型。在更广泛的背景下，这些“生物扩散”也可以被视为随机扩散和趋向性（即定向运动）组合的特殊情况。趋化性是另一个例子。这些极大地丰富了我们的扩散相关模型——从1952年著名的A.图灵的“扩散驱动的不稳定性”到最近发现的“扩散诱导的灭绝”和“扩散诱导的爆炸”——因此增强了我们对这些生物和物理现象建模的能力。本提案中的研究将以数学上严谨的方式，为理解各种类型的“生物扩散”（即随机、排斥和吸引）以及“的士”（即正性和负性）的集中/分离效应提供第一步。

9705639 Ni Professor Ni plans to continue his research in investigating various diffusion related mechanisms mathematically. A thorough understanding of those phenomena and effects of diffusions will enhance our ability in modelling more complicated or realistic situations in applied sciences. It seems that substantial progress has been made in the last ten years, and the stage is set for even more exciting mathematical breakthroughs, especially in diffusion related pattern formation. In this project, Professor Ni will study the following three closely related aspects: (I) Systems of semilinear diffusion equation (II) Systems of self-diffusion and cross-diffusion equations (III) Spike-layers (point condensation solutions) in diffusion/cross-diffusion systems. The purpose of this project is not only to separately develop these directions further, but to synthesize the results obtained and to build a theory to give a systematic point of view in understanding those "concentration" phenomena in differential equations. In an attempt to understand segregation phenomena in population dynamics, in 1979 several Japanese scientists at Kyoto University incorporated the notions of "self-diffusion" and "cross- diffusion" into the classical Lotka-Volterra model for the dynamics of two competing species. In a slightly broader context, these "bio-diffusions" could also be viewed as special cases of combinations of random diffusion and taxis (i.e. directed movements). Chemotaxis would be another such example. These have greatly enriched our diffusion-related models -- from the celebrated "diffusion-driven instability" of A. Turing in 1952 to recently discovered "diffusion-induced extinction" and "diffusion-induced blowup" -- and have therefore enhanced our ability in modelling those biological and physical phenomena. The research in this proposal would provide a first step toward the understanding, in a mathematically rigorous manner, of t he concentration/segregation effect of the various types of "bio-diffusions" (namely, random, repulsive, and attractive) as well as "taxis" (namely, positive and negative).