交叉扩散项对两类非线性偏微分方程组解的结构的影响研究

The nonlinear partial differential equations with cross-diffusion are the hotspot and also the nodus in the field of partial differential equations. This project will study two partial differential equations with cross-diffusion, which have a wide range of applications on ecology and pathology, such as pest control, embryonic development, cancer metastasis. The first is the dynamic model of two interacting populations, focused on the global well-posedness of solutions; the second is the chemotaxis fluid model of aerobic bacteria, concerned on the large time behaviour of solutions. Through the known works and the research content of the project, we will analyse the effect of cross-diffusion on the structure of solutions to two nonlinear partial differential equations compared with the corresponding equations without cross-diffusion. Due to the difficulties caused by strong nonlinearity and tight coupling, our research is a challenging work. We plan to set new variables and decompose high and low frequency, and so on. We also try to seek new methods and new techniques, which shall rich and perfect the mathematical theory of the nonlinear partial differential equations with cross-diffusion.

带交叉扩散项的非线性偏微分方程组是近年来偏微分方程领域研究的热点和难点问题。本项目将研究两类带交叉扩散项的偏微分方程组，它们在生态学、病理学等领域上都有广泛的应用，比如病虫害防治、胚胎发育、癌细胞转移等。第一类是两个相互作用种群迁移的动力学方程组，本项目考虑该方程组解的整体适定性；第二类是耗氧细菌种群演化的趋化流体方程组，考虑该方程组解的大时间性态。对这两类方程组，通过已有的工作和本项目计划研究的内容与对应的不带交叉扩散项的方程组作对比，分析交叉扩散项对解的结构的影响。由于交叉扩散的强非线性和耦合性会引起一些本质困难，对它们的研究是一项具有挑战性的工作，我们拟设立新变量和采用高低频分解等，同时通过本项目的研究试图探索新的方法和技巧，从而丰富和完善带交叉扩散项的非线性偏微分方程组的数学理论。

带交叉扩散项的非线性偏微分组是近年来偏微分方程领域研究的热点和难点问题。本项目计划研究两类带交叉扩散项的偏微分方程组。为此，首先，围绕可压的磁微极流模型，我们建立了强解的整体适定性和大时间行为，以及小能量弱解的整体存在性和解的正则性。其次，我们利用不可压Navier–Stokes–Maxwell方程组推导出了Hall-MHD方程组。最后，对基于PDE的二阶多智能体系统分别提出了没有时延现象和有时延现象两种情况下的编队控制协议。

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