关于两类动理学-流体耦合方程组相关问题的研究

This project will mainly study the mathematical theory about the following two types of kinetic-fluid coupled system: particle-fluid equations and radiation hydrodynamic equations. Now, the researches on both mathematical theory and scientific calculation about these two types of system have been the general interesting topics in complex system. Because the structure of these systems are hard, the mathematical theory of the related problems are beyond completeness, it requires further study. This project focuses on the well-posed, the time decay as well as the pointwise estimates of the solutions to these two coupled system and some related approximation models via the modern mathematical tools such as the energy estimates, the pseudo-differential operator besides the entropy-entropy dissipation method. Based on the well-posed of these systems, we also study the approximate theory such as the diffusion limit, the low Mach limit and the high-field limit. On the one hand, the researches in this project could help us understand the structure of the mixing system deeply and do research on the properties of the solutions. It is believed that the researches in this project would make a development in the mathematical theory on these two coupled systems. On the other hand, we derive some macroscopic equations through the approximate analysis and show the rigorous relationship between the original system and the limit one. It would provide the theoretical fundamental on the scientific calculation and mathematical simulation about the two types of coupled equations.

本项目主要探讨粒子-流体方程组和辐射流体力学方程组这两类动理学-流体耦合系统的数学理论。目前关于这两类系统的数学理论研究和科学计算已经成为复杂系统研究的热点，但由于系统结构的复杂性，相关问题的数学理论还不完善，有待进一步的研究。项目将应用能量方法、拟微分算子、熵-熵耗散等一些现代研究工具来讨论这两类系统及近似方程解的适定性、时间衰减、逐点估计，并在此基础上建立解的扩散极限、低Mach极限、高场极限等渐进理论。本项目的开展一方面有助于理解复杂系统的结构，研究解的性态，从而推进这两类耦合系统数学理论的发展；另一方面通过渐进分析推导宏观的近似数学模型，证明原系统与极限方程之间的严格渐进关系，为这两类耦合方程组的科学计算和数学模拟提供理论依据。

项目《关于两类动理学-流体耦合方程组相关问题的研究》主要关注动理学方程、辐射流体力学方程以及一些相关流体力学方程的数学理论与应用。主要内容有：动理学及耦合系统的适定性、解的大时间行为、非相对论极限；Euler-Poisson方程、Green–Naghdi 方程的长波极限, Navier-Stokes-Poisson方程的零质量极限；辐射流体力学近似方程的适定性、解的时间衰减及基本波的渐近稳定性；Navier-Stokes-Poisson方程的零质量极限以及Boussinesq方程的色散和扩散性质等。特别是在不同参数范围内，通过尺度变换形式推导出简单方程，并利用数学工具给出极限过程的严格数学论证，建立不同方程之间的联系，对偏微分方程数学理论和数值计算都有意义。项目组成员还对一些具有应用背景的方程进行数值模拟，推动方程的理论与实际应用的结合。项目执行期间，共发表22篇标注本项目号文章,这些论文发表在Adv. Math., J. Funct. Anal., Math. Mod. Meth. Appl.S., J. Differ. Equations等国际顶级刊物, 还有多篇文章已经整理完成并投稿。在此期间, 申请人组织了多次动理学方程研究的专题研讨会,吸引了大批学者参加,促进了学术交流与合作, 实现了项目预定目标.