Global Existence of Axisymmetric Pathwise Solutions for Stochastic Three-Dimensional Axisymmetric Navier-Stokes Equations

Abstract

A stochastic three-dimensional Navier-Stokes system with the axisymmetric initial data and white noise is studied. It is shown that if the swirl component of the initial velocity field and the white noise are sufficiently small, then the axisymmetric pathwise solution is global in probability. Moreover, in the absence of the swirl, the pathwise axisymmetric solution is global almost surely.