A probabilistic approach to the Navier-Stokes equations

该项目致力于利用概率技术研究Navier-Stokes方程。利用随机拉格朗日方法，我们将研究有界域上的Navier-Stokes方程的无粘极限。概率方法将允许我们解决新的问题(例如，从边界附近开始的粒子到达区域内部的概率的渐近极限)。该项目还将研究如下意义下的概率整体存在定理：找到一个期望是(确定性)Navier-Stokes方程强解的过程，并证明对于任意初始数据，该过程的爆破时间为无穷大且具有非零概率。这项研究的第二个目的是研究与不可压缩流动相关的非局部、非线性偏微分方程组，这些偏微分方程组促进了热点的产生。该项目将使用概率技术研究流体动力学的数学方面。一个根本的、悬而未决的问题将是边界层和零粘度极限的分离。这对于研究飞机机翼周围气流的确定等问题具有重要的实用价值。其他研究的问题是概率整体存在问题。虽然控制不可压缩流体演化的方程的解始终存在的问题尚未解决，但这个项目将解决这个问题的一个概率类比。也就是说，人们能否找到一个标准，允许人们确定某些流动的实现是动荡的，并阻止它们，并以非零的概率证明其余的“永远存在”？这个项目的第二个目标是研究某些特殊的搅拌方法，使液体保持热(即，什么是搅拌咖啡的最好方法，使它尽可能长时间热)；这适用于燃烧过程和化学反应。

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.