Higher dimension cross diffusion systems

本项目研究强耦合拟线性抛物型方程组边值问题解的性质。单个抛物方程解的许多著名性质的证明并不延伸到耦合系统，并且在这些方程的数值模拟中观察到了一些有趣的新现象。一个特别的兴趣是在证明这些系统的解决方案的规律性。对于两个方程的系统，最近已经得到了一些结果，我们将研究具有两个以上分量的系统。另一个目标是研究具有某些退化的强耦合抛物系统的长时间动力学和共存性。这类系统出现在多孔介质中的流体流动和物质混合问题中。物种和颗粒在它们的栖息地中移动或扩散，并相互作用。交叉扩散利用直接环境的信息研究物种/粒子的运动。我们将研究一些类别的交叉扩散系统，在建模化学，生态和机械应用中出现大量的变量。这一目标的进展将需要开发新的数学工具和方法，并有助于理解生命问题，如相互作用的群体是否以及如何持续存在。那就是生存和避免灭绝。该项目的成功完成将代表着在理解扩散策略（细胞运动，趋化性等）的作用方面迈出了重要的一步。在某些生态和生物应用中的竞争能力。

This project is to study the properties of solutions of boundary value problems for strongly coupled quasilinear parabolic systems of equations. The proofs of many of the well-known properties of solutions of a single parabolic equation do not extend to coupled systems and some interesting new phenomena have been observed in numerical simulations of these equations. A particular interest is in proving the regularity of solutions of these systems. Some results have recently been attained for systems of two equations and we will investigate systems with more than two components. Another goal is to investigate long time dynamics and coexistence for strongly coupled parabolic systems with certain degeneracies. Such systems arise in fluid flow in porous media, and material mixing problems.Species and particles move, or diffuse, and interact with each other in their habitats. Cross diffusion studies the motion of species/particles using information about the immediate environment. We will study some classes of cross diffusion systems with a large number of variables that arise in modeling chemical, ecological and mechanical applications. Progress in this objective will require the development of new mathematical tools and methods, and also help to understand life questions such as whether and how a community of interacting populations can persist. That is survive and avoid extinction. The successful completion of this project will represent a significant step forward in the understanding of the roles of dispersal strategies (cell motilities, chemotaxis, etc.) and competitive abilities in certain ecological and biological applications.