A Stochastic Langrangian approach to non-linear transport equations

这一数学领域的方法粘性输运方程的一个嘈杂的逆流动图。这是一个自然的推广的方法的特征（用于无粘问题）粘性问题，通过添加噪音的粒子轨迹和平均。本研究的主要重点是应用这些技术的Navier-Stokes和Boussinesq方程。Navier-Stokes方程可以在上述框架中使用无粘Webber公式优雅地表示，其基本上将Navier-Stokes方程表示为欧拉方程加上布朗运动的平均值。我们考虑了各种各样的应用：数值计算Navier-Stokes方程解的（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，分析估计和弱解，Boussinesq方程中的波前传播。这项研究是基于粘性不可压缩流体可以明确表示为无粘性流体加上布朗运动的平均值。在这个框架内的流体动力学的几个方面的研究提出。一个重要的应用是实施一种数值方法来计算湍流环境中的流体流动（例如机翼周围的气流或喷气发动机中的气流）。一个相关的应用是使用这种方法来找到流体流动的“湍流模型”：即，流体的速度场被表示为缓慢变化的“平均场”（这是容易和廉价的计算），加上快速波动的“噪声”部分，这是使用随机方法建模。另一个应用是研究边界层对无粘极限内流的影响。还提出了燃烧和火焰传播的研究。

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.