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.
