Mathematical Sciences: Diffusion, Cross-Diffusion and Spike Layers

9705639倪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。彻底了解这些现象和扩散的影响将提高我们对应用科学中更复杂或更现实的情况进行建模的能力。似乎在过去的十年里已经取得了实质性的进展，并为更令人兴奋的数学突破奠定了基础，特别是在与扩散相关的图案形成方面。在这个项目中，倪教授将研究以下三个密切相关的方面：(I)半线性扩散方程组(Ii)自扩散方程组和交叉扩散方程组(Iii)扩散/交叉扩散系统中的尖峰层(点凝聚解)。这个项目的目的不仅是为了进一步单独地发展这两个方向，而且是为了综合已有的结果，并建立一个理论来为理解微分方程中的“集中”现象提供一个系统的观点。为了理解种群动态中的隔离现象，1979年，京都大学的几位日本科学家在经典的Lotka-Volterra模型中加入了“自扩散”和“交叉扩散”的概念，以研究两个相互竞争的物种的动态。在稍微更广泛的背景下，这些“生物扩散”也可以被视为随机扩散和趋势性的组合(即定向移动)的特例。趋化性将是另一个这样的例子。这些极大地丰富了我们与扩散有关的模型--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”--因此提高了我们模拟这些生物和物理现象的能力。这项提案中的研究将为以严格的数学方式理解各种类型的“生物扩散”(即随机、排斥和吸引)以及“趋同”(即积极和消极)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).