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

A.项目技术说明。在这个项目中，我们研究非线性抛物型和双曲型偏微分方程和相互作用粒子系统的问题，共同的潜在主题是多尺度的存在和相互关系。我们讨论以下主题：(a)与多尺度粒子系统相互作用，在中观水平上产生离散速度模型，在宏观水平上产生双曲守恒定律或非线性扩散；(b)在松弛机制存在的情况下，双曲守恒定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数值方案；(d)伊辛模型对几何演化的宏观极限中的蒙特卡罗方法和随机效应。除了项目的分析方面外，(c)的目标之一是通过松弛模型为界面跟踪应用开发新的计算工具；此外，在(d)中，采用了基于随机Ising模型的数值实验，试图从微观角度理解随机扰动对界面演化的影响。b .项目的非技术描述。涉及多个相互作用尺度的现象在材料科学、化学工程、生物学、大气与环境科学、通信网络和经济学等领域广泛存在。一个典型的例子是，从材料的微观分子甚至原子结构推导出材料的宏观特性，例如导电性。这些问题给科学家和工程师带来的挑战在很多层面上都是巨大的：在实验、建模、模型的数学分析、有效算法的构建、模型的仿真和验证。多尺度问题固有的数学复杂性和我们迄今为止对其本质的有限理解，要求对实际问题和（仔细地）简化问题进行系统和详细的分析，以更清楚地说明所研究现象的特定方面。提出的研究主要涉及多尺度过程中一些范式的数学分析，并涉及数学理论所建议的新计算工具的发展。这项工作的动机源于材料科学、化学工程应用中的相变问题，如吸附过程、水波和气体动力学。

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.