A probabilistic approach to the Navier-Stokes equations

纳维-斯托克斯方程的概率方法

• 批准号: 1007914
• 负责人: Gautam Iyer
• 金额: $ 16.51万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007914&HistoricalAwards=false
• 关键词: probabilistic; approach; Navier; Stokes; equations

The project is devoted to a study of the Navier-Stokes equations using probabilistic techniques. Using stochastic Lagrangian techniques we will study the inviscid limit of the Navier Stokes equations in bounded domains. The probabilistic approach will allow us to address new questions (e.g. the asymptotic limit of the probability that a particle starting close to the boundary arrives in the interior of the domain). The project will also study probabilistic global existence theorems in the following sense: find a process whose expectation is a strong solution to the (deterministic) Navier-Stokes equations, and prove that for arbitrary initial data, the blow up time of the process is infinity with non-zero probability. A secondary aim of this research is to study the nonlocal, nonlinear PDEs associated with incompressible flows that promote the creation of hot spots.The project will study mathematical aspects of fluid dynamics using probabilistic techniques. One fundamental, unresolved question addressed will be the separation of the boundary layer and the zero viscosity limit. This is of great practical interest in studying problems such as the determination of air flow about an airplane wing. Other problems studied are probabilistic global existence questions. While the question of existence for all time of solutions of the equations governing the evolution of incompressible fluids is unresolved, this project will address a probabilistic analogue of this question. Namely, can one find a criterion that allows one to decide that certain realizations of the flow are turbulent and stop them, and show that the remainder "live forever" with nonzero probability? A secondary aim of this project is to study certain peculiar stirring methods that keep the fluid hot (i.e. what is the best way to stir your coffee so that it is hot for as long as possible); this has applications to combustion processes, and chemical reactions.

该项目用于研究使用概率技术对Navier-Stokes方程的研究。使用随机拉格朗日技术，我们将研究有限域中Navier Stokes方程的无关极限。概率方法将使我们能够解决新问题（例如，粒子启动到边界接近边界的概率的渐近极限到达域内的内部）。该项目还将从以下意义上研究概率的全球存在定理：找到一个过程，其期望是（确定性的）navier-stokes方程的强大解决方案，并证明对于任意初始数据，该过程的爆炸时间是无限的，具有非零概率。这项研究的次要目的是研究与不可压缩流相关的非本地的非线性PDE，这些流量促进了热点的创建。该项目将使用概率技术研究流体动力学的数学方面。一个基本的未解决的问题将是边界层的分离和零粘度极限。这对于研究问题，例如确定飞机翼的空气流量，这是很大的兴趣。研究的其他问题是概率全球存在问题。尽管尚未解决有关不可压缩流体演变的方程式解决方案解决方案的存在问题，但该项目将解决这个问题的概率类似物。也就是说，一个人可以找到一个允许人们确定某些流动性的标准是动荡的，并阻止它们，并证明其余的“永远活着”具有非零概率？该项目的次要目的是研究某些使液体热的特殊搅拌方法（即，搅拌咖啡的最佳方法是什么是最佳的咖啡方法，以使其尽可能长时间热）；这具有燃烧过程和化学反应的应用。