A probabilistic representation of solutions of the incompressible Navier-Stokes equations in R3

2004/12/01 by Mina Ossiander, M. Ossiander, Ossiander, M.
Engineering · Mathematics · #Computational Fluid Dynamics and Aerodynamics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.math/0412034

29 pages, 1 figure, to appear in Probability Theory and Related Fields

arxiv created 2004/12/01 · arxiv updated 2009/12/01

Abstract

A new probabilistic representation is presented for solutions of the incompressible Navier-Stokes equations in 3 dimensions with given forcing and initial velocity. This representation expresses solutions as scaled conditional expectations of functionals of a Markov process indexed by the nodes of a binary tree. It gives existence and uniqueness of weak solutions for all time under relatively simple conditions on the forcing and initial data. These conditions involve comparison of the forcing and initial data with majorizing kernels.

Citations

• About the regularized Navier--Stokes equations

Related

• Probabilistic representation for mild solution of the Navier-Stokes equations
• Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier--Stokes equations: existence and non-uniqueness
• Global existence and non-uniqueness for 3D Navier--Stokes equations with space-time white noise
• Nonuniqueness in law for stochastic hypodissipative Navier-Stokes equations
• A probabilistic representation for the vorticity of a 3D viscous fluid and for general systems of parabolic equations