A Stochastic Langrangian approach to non-linear transport equations

这一数学领域通过有噪声的逆流图来逼近粘性输运方程。这是特征线法(用于无粘问题)对粘性问题的自然推广，通过在粒子轨迹中添加噪声和平均来实现。本研究的重点是将这些技术应用于Navier-Stokes方程和Boussinesq方程。在上述框架中，Navier-Stokes方程可以使用无粘性Webber公式来表示，该公式实质上将Navier-Stokes方程表示为Euler方程加上布朗运动的平均值。我们有各种各样的应用：计算Navier-Stokes方程解的数值(随机)方法，研究障碍物和边界对粘性流体流动的影响，不可压缩流体的湍流模型，解析估计和弱解，以及Boussinesq方程中的前沿传播。这项研究基于这样的框架，即粘性不可压缩流体可以显式地表示为无粘流体加布朗运动的平均值。在此框架下，对流体动力学的几个方面进行了研究。一个重要的应用是实现一种数值方法来计算湍流环境中的流体流动(例如，飞机机翼周围的空气流动，或喷气发动机中的空气流动)。一个相关的应用是使用这种方法来寻找流体流动的“湍流模型”：即，流体的速度场被表示为缓慢变化的“平均场”(计算简单且成本低廉)，加上快速波动的“噪声”部分，该部分是使用随机方法建模的。另一个应用是研究边界层对无粘边界内流动的影响。并对燃烧和火焰传播进行了研究。

This area of mathematics approaches viscous transport equations in terms of a noisy inverse flow map. This is a natural generalization of the method of characteristics (used for inviscid problems) to viscous problems by adding noise to particle trajectories and averaging. The main focus of this research is to apply these techniques to the Navier-Stokes and Boussinesq equations. The Navier-Stokes equations can be elegantly formulated in the above framework using the inviscid Webber formula, which essentially represents the Navier-Stokes equations as the average of the Euler equations plus Brownian motion. We have a wide variety of applications in mind: A numerical (stochastic) method to compute solutions to the Navier-Stokes equations, a study of the effect of obstacles and boundaries on viscous fluid flows, turbulence models for incompressible fluids, analytical estimates and weak solutions, and front propagation in the Boussinesq equations.This research is based on the framework in which a viscous incompressible fluid can be explicitly represented as the average of an inviscid fluid plus Brownian motion. A study of several aspects of fluid dynamics in this framework is proposed. One important application is the implementation of a numerical method to compute fluid flows in turbulent settings (for instance air flow around an airplane wing, or in a jet engine). A related application is to use this method to find a "turbulence model" for fluid flows: namely, the velocity field of the fluid is expressed as a slowly varying "mean field" (which is easy and inexpensive to compute), plus a rapid fluctuating "noisy" part, which is modeled using stochastic methods. Another application is to study boundary layer effects on the interior flow in the inviscid limit. A study of burning and flame propagation is also proposed.