Nonlinear Partial Differential Equations in Continuum Mechanics

连续介质力学中的非线性偏微分方程

• 批准号: 9971978
• 负责人: Zhouping Xin
• 金额: $ 8.74万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971978&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations; Continuum

This award will support research on nonlinear evolution partial differentialequations arising from continuum mechanics. Examples include the Euler and Navier-Stokes equations for compressible and incompressible inviscid and viscous fluids, the equations of electro-magneto field dynamics for electrically conducting compressible viscous fluids, the equations of elasticity, nonlinear Boltzmann type equations in kinetic theory of rarefied gases, and equations for combustion, multiphase flows, and fluids with chemical reactions.The main objective is to gain better qualitative and quantitative understanding of the behavior of solutions, using mathematically rigorous analysis and numerical simulations. The focus will be on understanding the effects of small scale dissipations andrelaxations on the large scale fluid motions; development of singularitiesand their stuctures from smooth flows; nonlinear stability or instability and interactions for basic linear and nonlinear waves; design and analysis ofhigh resolution numerical methods for large scale calculations of discontinous flows;fluid-dynamic limit problems for various kinetic models; and problems in multi-spacedimensions and in thermo-nonequilibrium. One of the main ideas is to overcome the difficulties that are encountered at the macroscopic level bydesigning and analyzing approximate models which are physically more fundamental and mathematically tractable. Many interesting phenomena in the mechanics of continua can be described by mathematical models that involve nonlinear partial differential equations. A better understanding of these models has important applications in many sciences such as mechanics, turbulence theory, geophysics, meteorology, aeronautics, chemical engineering, etc. This award will support work on these fundamental models.The goal is to gain better qualitative understanding and to improve the foundationsfor effective and accurate numerical simulations.

该奖项将支持研究非线性演化偏微分方程所产生的连续介质力学。 例子包括可压缩和不可压缩的无粘和粘性流体的欧拉和纳维尔-斯托克斯方程，导电可压缩粘性流体的电磁场动力学方程，弹性方程，稀薄气体动力学理论中的非线性玻尔兹曼方程，以及燃烧、多相流、和具有化学反应的流体。主要目标是使用数学上严格的分析和数值模拟，更好地定性和定量地了解溶液的行为。 重点是了解小尺度耗散和驰豫对大尺度流体运动的影响;光滑流中奇异性及其结构的发展;基本线性和非线性波的非线性稳定性或不稳定性以及相互作用;设计和分析用于大规模计算的高分辨率数值方法不连续流;各种动力学模型的流体动力学极限问题;以及多维空间和热不平衡问题。 主要思想之一是通过设计和分析物理上更基本和数学上更易处理的近似模型来克服在宏观层面上遇到的困难。 连续介质力学中许多有趣的现象都可以用包含非线性偏微分方程的数学模型来描述。更好地理解这些模型在许多科学中有重要的应用，如力学，湍流理论，物理学，气象学，航空学，化学工程等。该奖项将支持这些基础模型的工作。目标是获得更好的定性理解，并提高有效和准确的数值模拟的基础。