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

该项目将利用随机微积分和概率极限理论的力量，利用新的概率方法，发展流体运动理论，如纳维-斯托克斯方程所体现的理论。虽然Navier-Stokes方程本质上是确定性的，但在这项工作中使用的方法将建立在将方程表示为潜在分支随机游走的函数的基础上。这种表示，最近由法国的LeJan和Sznitman发现，显然是Navier-Stokes方程结构的内在特征。虽然这不是第一次尝试使用随机方法来研究与Navier-Stokes方程相关的流动，但它确实代表了一个全新的方向，有可能超越现有的许多理论。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、多尺度指数和奇点、粘度、均匀性、各向同性和旋转加速度、静止流动和长时间演化的作用。纳维-斯托克斯方程描述了控制各种形式的空气、水、油等流体运动的基本物理。因此，这些方程通过模拟各种流体流动，从大气和海洋环流到地球表面下的水流，在科学和工程中发挥着重要作用。提高对这些方程及其解决方案的理解对于从跟踪气候变化和地球环境中污染物扩散到更稳定的航空航天和海洋船舶设计等应用至关重要。这些方程固有的非线性使得只有最简单的流动才能得到显式解。因此，在所有物理长度尺度上对这些方程进行更全面的理解，是当代数学物理学中最重要的突出问题之一

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