Collaborative Research: Stochastic and Multiscale Structure Associated with the Navier Stokes Equations

合作研究：与纳维斯托克斯方程相关的随机和多尺度结构

• 批准号: 0244485
• 负责人: Rabindra Bhattacharya
• 金额: $ 2.36万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244485&HistoricalAwards=false
• 关键词: Collaborative; Research; Stochastic; Multiscale; Structure

This project will develop the theory of the motion of fluids as embodied by the Navier-Stokes equations using new probabilistic methods that exploit the power of stochastic calculus and probabilistic limit theory. Although the Navier-Stokes equations are essentially deterministic, the approach used in this work will build on a representation of the equations as a functional of an underlying branching random walk. This representation, which was recently discovered by LeJan and Sznitman in France, is clearly intrinsic to the structure of the Navier-Stokes equations. While this is not the first attempt to use stochastic methods in connection with the flows associated with the Navier-Stokes equations, it does represent an entirely new direction which has the potential to transcend much of existing theory. Specific problems considered in this proposal seek to provide a better understanding of the role of spatial dimensions, boundary conditions, multi-scaling exponents and singularities, viscosity, homogeneity, isotropy and rotational accelerations, stationary flows and long-time evolution. The Navier-Stokes equations describe the basic physics governing the motion of fluid in its various forms of air, water, oil, etc. As such these equations play a fundamental role in science and engineering through the modeling of all varieties of fluid flow, from atmospheric and oceanic circulation to the flow of water beneath the earth's surface. Improved understanding of these equations and their solutions is essential to applications which range from tracking climate change and dispersion of contaminants in the Earth's environment, to more stable aerospace and sea vessel designs. The nonlinearity inherent in these equations makes explicit solutions possible only for the simplest of flows. Consequently the development of a more complete understanding of these equations at all physical length scales ranks among the most important outstanding problems of contemporary mathematical physics

该项目将使用新的概率方法来开发流体运动的运动理论，这些方法由Navier-Stokes方程所体现，从而利用随机演算的力量和概率极限理论。尽管Navier-Stokes方程本质上是确定性的，但本工作中使用的方法将以方程式为基础分支随机步行的功能建立。这种表示形式最近由法国的Lejan和Sznitman发现，显然是Navier-Stokes方程的结构。尽管这不是与与Navier-Stokes方程相关的流进行随机方法的首次尝试，但它确实代表了一个全新的方向，有可能超越现有理论。本提案中考虑的具体问题旨在更好地理解空间维度，边界条件，多尺度指数和奇异性，粘度，同质性，各向同性和旋转加速度，固定流量和长期进化的作用。 Navier-Stokes方程描述了液体以各种形式的空气，水，油等的运动的基本物理学。因此，通过建模各种流体流量，从大气和海洋循环到地面以下水流的流动，这些方程在科学和工程中起着基本作用。对这些方程式及其解决方案的理解提高了，对于从跟踪气候变化和地球环境中污染物的分散到更稳定的航空航天和海船设计的应用至关重要。这些方程式固有的非线性使得明确的解决方案仅对最简单的流量而成为可能。因此，在所有物理长度尺度上，对这些方程式的更全面理解是当代数学物理学最重要的问题之一