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

这个项目将使用新的概率方法，利用随机微积分和概率极限理论的力量，发展体现在Navier-Stokes方程中的流体运动理论。虽然Navier-Stokes方程本质上是确定性的，但本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动的泛函的基础上。这种表象是由法国的Lejan和Sznitman最近发现的，它显然是Navier-Stokes方程结构的内在特征。虽然这并不是第一次尝试将随机方法与与Navier-Stokes方程相关的流动联系在一起，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。这一建议中考虑的具体问题力求更好地理解空间维度、边界条件、多尺度指数和奇点、粘性、均质性、各向同性和旋转加速、定常流动和长期演变的作用。Navier-Stokes方程描述了控制空气、水、油等各种形式的流体运动的基本物理。因此，这些方程通过对从大气和海洋环流到地球表面下的水流动的所有类型的流体流动的建模，在科学和工程中发挥着基本的作用。更好地了解这些方程及其解对于从跟踪气候变化和污染物在地球环境中的扩散到更稳定的航空航天和海船设计的各种应用至关重要。这些方程固有的非线性使得只有最简单的流动才有可能得到显式解。因此，对这些方程在所有物理长度尺度上的更完整的理解的发展是当代数学物理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