动力学模型与扩散

Classical models for biomath usually came from the empirical diffusion type partial diffusion equations, where the physical constants are determined by experimental data. As the fast development of life science in the last few years, many biological phenomina couldn't be simply described by these classes of diffusion models. Applied mathematicians proposed a new approach based on microscopic kinetic equations, in which the main reseach idea was similar to what have been used in mathematical physics for many years. They treat biology agents as particles, derived the complex system according to their moving stratergy. Then by studying the mean field limit of this ODE system, they were able to get a mesoscopic kinetic equation, an equation from statistical physics. Furthermore, by studying different kinds of singular limit, one can derive fluiddynemic type PDE system. This project is intended to study a few definite problems within these limiting procedures. Namely, includes existence, uniqueness and large time behavior of measure valued solution on the kinetic system with non-Lipschitz potential. The swarming phenomina and mean field limit of microscopic PTW and PTWA model. Wellposedness of the fluid dynamic model derived from PTWA. The criteria of global existence and finite time blow up on multi-dimensional degenerate Keller-Segel system. We hope that our research in this project will solve some key problems in the newly developed way of study biomath, and help in the development of it.

传统的生物数学模型来自于经验式的具有扩散效应的偏微分方程组，其参数设置来源于实验数据。近年来随着生命科学的高速发展，很多生物现象不能简单的由此类传统模型描述。应用数学家们提出了一类基于微观动力学模型的研究方法，其研究思路类似于数学物理，将生物个体看作是粒子，给出多粒子的运动学方程，然后通过寻求其平均场极限得到动力学的介观方程,一种统计力学方程。进而通过考虑各种奇异极限导出具有流体力学方程组性质的偏微分方程组。本项目旨在研究整个极限过程的几个特定问题，包括微观动力学模型在非Lipschitz连续势的情形，测度解的存在唯一性以及其大时间行为，PTW和PTWA模型的蜂拥现象和平均场极限问题，由PTWA导出的流体力学类方程的适定性问题，退化Keller-Segel方程组解的整体存在与爆破问题。期望能够解决一些这个近几年来研究生物数学的新方向中的关键性问题。

本项目就近年来生命科学及数学生物学领域提出的一大类数学模型进行了系统的理论分析，其中包括描述细胞运动的多粒子运动学模型（有界Lip连续的作用势以及具有奇性的相互作用势情况），由其导出的介观Vlasov型的模型以及进而经过奇异极限导出的宏观扩散模型。特别需要指出的是，本项目一方面解决了由具有cut-off的奇异相互作用势的粒子运动学模型到非cut-off作用势的Vlasov型模型平均场极限的严格证明，另一方面对于相应的扩散模型（包括带驱化效应的Keller-Segel方程组以及带有Fisher-KPP反应项的扩散方程）给出了完整的解的适定性理论，尤其是给出了高维情形关于解的存在与爆破的严格划分标准。项目的结果大部分已经在颇有影响力的学术期刊发表，为生物数学中动力学模型方面的发展提供理论基础。