Mathematical Analysis of Moment Systems Derived from Kinetic Equations

偏微分方程是现代科学中不可缺少的工具。许多物理现象，如流体运动、声音传播和热传递，都可以用某些类型的偏微分方程组来模拟。偏微分方程组的理论研究对于我们理解复杂系统的基本性质或行为是至关重要的，我的研究主要集中在对偏微分方程组的分析。这项研究计划的主要主题是研究与一种称为动力学方程的特殊类型的偏微分方程有关的模型。动力学方程是模拟大系统中粒子密度分布演化的经典工具。动力学方程的传统应用领域包括核反应堆、半导体和等离子体中的气体动力学和传输现象。近年来，动力学理论在生物学、行为科学和社会网络等其他科学分支中的应用也不断涌现。在实践中，当一个系统中涉及太多粒子时，直接模拟动力学方程往往是困难的。解决这一困难的典型方法是构建模型来近似原始的动力学方程。在这个提案中，我将研究一族被称为矩系统的近似族。这些系统将整个粒子分布减少到与分布函数相关的有限多个平均量。我在这个程序中的主要目标是构造和分析适当的力矩系统，以便在逼近基本的动力学方程时能够提供高精度。此外，这些矩系统将作为经典流体方程的推广，如纳维尔-斯托克斯系统。因此，在经典流体方程不再适用的情况下，它们将是合适的模型。该程序将为捕捉多尺度现象的建模和计算提供便利。

Partial differential equations (PDEs) are indispensable tools in modern science. Many physical phenomena such as fluid motion, sound propagation, and heat transfer can all be modeled by certain types of PDEs. Theoretical investigation of PDEs is crucial for us to understand the fundamental properties or behaviors of complex systems.My research concentrates on analysis of PDEs. The main topic of this research program is to study models related to a particular type of PDEs called kinetic equations. Kinetic equations are classical tools to model the evolution of density distributions of particles in large systems. Traditional application areas of kinetic equations include gas dynamics and transport phenomena in nuclear reactors, semiconductors, and plasmas. More recent years have also seen emerging applications of kinetic theory to other branches of science, such as biology, behavioral science, and social networks.In practice when there are too many particles involved in a system, it is often difficult to directly simulate kinetic equations. A typical way to resolve this difficulty is to construct models to approximate the original kinetic equation. In this proposal I will study a family of approximations called moment systems. These systems reduce the full particle distribution to only finitely many averaged quantities related to the distribution function. My main goal in this program is to construct and analyze proper moment systems which can give high accuracy when approximating the underlying kinetic equations. In addition, these moment systems will serve as extensions of classical fluid equations such as the Navier-Stokes system. Hence they will be appropriate models in the regimes for which classical fluid equations are no longer valid. This program will facilitate modeling and computations for capturing multiscale phenomena.