Convergence of Parareal Algorithms for PDEs with Fractional Laplacian and a Non-Constant Coefficient

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.

Auxiliary Equations Approach for the Stochastic Unsteady Navier--Stokes Equations with Additive Random Noise

“…The Martingale regularization method leads the simulation and analysis of the stochastic Navier-Stokes equations (1.1) to the stochastic Stokes equations (1.3) which are linear and the stochastic modified Navier-Stokes equations (1.4) with smoother noise. For some rigorous theoretical analyses of the stochastic Navier-Stokes equations, readers can refer to [2,4,5,7,11,13,17], etc. Some interesting topics of the stochastic partial differential equations (SPDEs) are further detailed in [1,6,8,14,15,18].…”