Mathematical Analysis of Moment Systems Derived from Kinetic Equations

由动力学方程导出的力矩系统的数学分析

• 批准号: 435842-2013
• 负责人: Sun, Weiran
• 金额: $ 1.09万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635599
• 关键词: Mathematical; Analysis; Moment; Systems; Derived

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 Hence they will be appropriate models in the regimes for which classical fluid equations are no longer valid.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.

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，声音传播和传热都可以通过某些类型的偏微分方程建模。偏微分方程的理论研究对于我们理解复杂系统的基本性质和行为至关重要，我的研究集中在偏微分方程的分析上。该研究计划的主要课题是研究与称为动力学方程的特定类型的偏微分方程相关的模型。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输运现象。近年来，动力学理论在生物学、行为科学和社交网络等其他学科中也有了新的应用。在实际应用中，当系统中涉及的粒子太多时，直接模拟动力学方程往往是困难的。解决这一难题的一个典型方法是建立模型来近似原动力学方程。在这个建议中，我将研究一族近似，称为矩系。这些系统将完整的粒子分布减少到与分布函数相关的仅1000个平均量。我在这个程序中的主要目标是构建和分析适当的时刻系统，可以提供高精度时，近似基本的动力学方程。此外，这些矩系统将作为经典流体方程，如Navier-Stokes系统的扩展。因此，他们将是适当的模式，在政权的经典流体方程不再有效。这个程序将有助于捕捉多尺度现象的建模和计算。