A stochastic Lagrangian formulation of the incompressible Navier -Stokes and related *transport equations
In this dissertation we derive a representation of the deterministic 3-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey a stochastic differential equation with the (deterministic) velocity of the fluid as the drift, and a constant diffusion coefficient. We then recover the fluid's velocity field by using the inviscid Weber formula and averaging out the noise. We use this idea to provide formulations of related non-linear transport equations, including the viscous Burgers equations, Camassa-Holm and reaction diffusion equations.
We use the formulation above to provide a self contained local existence proof of the Navier-Stokes equations in spatially periodic domains. We conclude by examining a stochastic Lagrangian model where the particle trajectories obey a SDE where the drift is computed from the flow map without averaging. We show that the average of the velocity obtained from this system is a random translate of the Euler equations, and a super-linear approximation to the velocity field of the Navier-Stokes equations.