A probabilistic approach to the Navier-Stokes equations

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

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.