Mathematical Structure in Fluid Mechanics

流体力学的数学结构

• 批准号: 0607953
• 负责人: Alexis Vasseur
• 金额: $ 12万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0607953&HistoricalAwards=false
• 关键词: Mathematical; Structure; Fluid; Mechanics

VasseurDms-0607953 The investigator carries out a qualitative study of partialdifferential equations arising as models in fluid mechanics andchemistry. Understanding the properties of these models is afundamental challenge both in mathematics and science. Theregularity and stability of solutions are issues strongly relatedto nontrivial physical phenomena like turbulence and phasetransitions that are complex multiscale problems. Analyzingmathematical models of these phenomena often requires thedevelopment of novel tools encompassing techniques from analysis,applied mathematics, and stochastics. This project, whichaddresses both theoretical and applied issues, spans fromconservation laws to Navier-Stokes equations, kinetic equations,and motion of particles in fluids. It requires tools from thetheories of nonlinear partial differential equations, asymptoticanalysis, probability, and stochastics. The discovery of mathematical structure in physical modelsis of great importance for a deeper understanding of the physicalphenomena themselves. It gives also some hints of the range ofvalidity of the models. In this project, the investigatordevelops and studies new models involving couplings between fluidmechanics and physics far from equilibrium. This field hasimportant applications in science and engineering. For example,it models pollutants in a river or combustion of gasoline in anengine. The applied and mathematical challenges stem from thecomplexity of the phenomena.

研究者对流体力学和化学中作为模型出现的偏微分方程进行定性研究。理解这些模型的性质是数学和科学的基本挑战。解的规则性和稳定性是与复杂多尺度问题的湍流和相变等重要物理现象密切相关的问题。分析这些现象的数学模型通常需要开发新的工具，包括分析、应用数学和随机学等技术。该项目涉及理论和应用问题，涵盖从守恒定律到Navier-Stokes方程、动力学方程和流体中粒子的运动。它需要非线性偏微分方程、渐近分析、概率和随机等理论的工具。在物理模型中发现数学结构对于更深入地理解物理现象本身具有重要意义。给出了模型的有效范围的一些提示。在这个项目中，研究者开发和研究了远离平衡的流体力学和物理之间的耦合的新模型。这个领域在科学和工程上有重要的应用。例如，它模拟河流中的污染物或发动机中汽油的燃烧。应用和数学上的挑战源于现象的复杂性。