A probabilistic approach to the Navier-Stokes equations

该项目致力于使用概率技术研究纳维尔-斯托克斯方程。利用随机拉格朗日方法，我们将研究有界区域中Navier-Stokes方程的无粘极限。概率方法将使我们能够解决新的问题（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）。该项目还将在以下意义上研究概率全局存在定理：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，该过程的爆破时间是无穷大，概率为非零。本研究的第二个目的是研究与不可压缩流相关的非局部非线性偏微分方程，这些偏微分方程促进了热点的产生。该项目将使用概率技术研究流体动力学的数学方面。一个基本的，未解决的问题将是边界层的分离和零粘度极限。这在研究诸如确定机翼周围的气流等问题时具有很大的实际意义。研究的其他问题是概率全局存在问题。虽然所有时间的不可压缩流体的演化方程的解的存在性问题尚未解决，这个项目将解决这个问题的概率模拟。也就是说，我们能否找到一个标准，让我们决定流动的某些实现是动荡的，并停止它们，并表明其余的“永远存在”的非零概率？这个项目的第二个目的是研究某些特殊的搅拌方法，使流体保持热（即什么是搅拌咖啡的最佳方法，使其尽可能长时间保持热）;这对燃烧过程和化学反应有应用。

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.