Multi-Scale Analysis for Nonlinear and Partial Differential Equations and Interacting Particle Systems

非线性和偏微分方程以及相互作用粒子系统的多尺度分析

• 批准号: 9801769
• 负责人: Markos Katsoulakis
• 金额: $ 15.5万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801769&HistoricalAwards=false
• 关键词: Multi; Scale; Analysis; Nonlinear; Partial

DMS-9801769 Markos ABSTRACT A. Technical description of the project. In this project we study problems in nonlinear parabolic and hyperbolic partial differential equations and interacting particle systems, with common underlying theme the presence and interrelation of multiple scales. We address the following topics: (a) interacting particle systems with multiple scales, giving rise to discrete velocity models at a mesocopic level, and to hyperbolic conservation laws or nonlinear diffusions, at a macroscopic one, (b) existence and stability of periodic and solitary waves for hyperbolic conservation laws in the presence of relaxation mechanisms, (c) relaxation models and related numerical schemes for front propagation problems, and (d) Monte-Carlo methods and random effects in the macroscopic limit of Ising models to geometric evolutions. In addition to the analytical aspects of the project, one of the objectives in (c) is the development of new computational tools, through relaxation models, for interface tracking applications; furthermore in (d), numerical experiments based on stochastic Ising models are employed, in an attempt to understand from a microscopic point of view the effect of random perturbations on interfacial evolutions. B.Non-technical description of the project. Phenomena involving multiple, interacting scales are widespread in areas as diverse as material science, chemical engineering, biology, atmospheric and environmental sciences, communication networks and economics. A typical example is the derivation of macroscopic properties of a material, for instance conductivity, from its microscopic-molecular or even atomic-structure. The challenges these problems present to scientists and engineers are formidable at many levels: in experiments, modelling, mathematical analysis of models, construction of effective algorithms, simulation and verification of the models. The inherent mathematical complexity of multiscale problems and our limited understanding of their nature so far, require a methodical and detailed analysis both of the actual problems, as well as of (carefully) simplified ones, that illustrate in a clearer way a particular aspect of the phenomenon under study. The proposed research addresses mainly the mathematical analysis of some paradigms in multiscale processes and touches upon the development of new computational tools, suggested by the mathematical theory. The motivation of the work stems from phase transitions problems in material science, chemical engineering applications such as adsorption processes and water waves and gas dynamics.

9801769 MARKOS摘要A.项目的技术描述。在这个项目中，我们研究了非线性抛物型和双曲型偏微分方程组和相互作用的粒子系统的问题，共同的基本主题是多个尺度的存在和相互关系。我们讨论以下问题：(A)多尺度相互作用的粒子系统，在介观水平上产生离散的速度模型，在宏观水平上引起双曲守恒律或非线性扩散，(B)存在松弛机制的双曲守恒律的周期波和孤波的存在性和稳定性，(C)前沿传播问题的松弛模型和相关的数值格式，以及(D)蒙特卡罗方法和伊辛模型宏观极限对几何演化的随机效应。除了项目的分析方面，(C)中的目标之一是开发新的计算工具，通过松弛模型，用于界面跟踪应用；此外，在(D)中，采用基于随机伊辛模型的数值实验，试图从微观角度了解随机扰动对界面演化的影响。B.非技术描述-项目描述。涉及多个相互作用的尺度的现象在材料科学、化学工程、生物学、大气和环境科学、通信网络和经济等不同领域广泛存在。一个典型的例子是从微观分子结构甚至原子结构推导出材料的宏观性质，例如导电性。这些问题对科学家和工程师提出的挑战在许多层面上都是艰巨的：在实验中，建模，模型的数学分析，有效算法的构建，模型的模拟和验证。多尺度问题内在的数学复杂性，以及我们迄今对其性质的有限理解，需要对实际问题以及(仔细地)简化的问题进行系统和详细的分析，以便更清楚地说明正在研究的现象的特定方面。拟议的研究主要针对多尺度过程中的一些范例的数学分析，并涉及到数学理论所提出的新的计算工具的开发。这项工作的动机来自材料科学中的相变问题，以及吸附过程、水波和气体动力学等化学工程应用。