Convergence of a sequence of solutions of the stochastic two-dimensional equations of second grade fluids

TL;DR

This paper proves that solutions of stochastic second grade fluid equations in two dimensions converge to solutions of stochastic Navier-Stokes equations as the stress modulus approaches zero, under certain conditions.

Contribution

It establishes the convergence of stochastic second grade fluid solutions to stochastic Navier-Stokes solutions in two dimensions as stress modulus tends to zero.

Findings

Solutions of stochastic second grade fluids converge to Navier-Stokes solutions

Convergence holds under suitable data conditions

Unique strong probabilistic solutions are involved

Abstract

We study the limit of the stochastic model for two dimensional second grade fluids subjected to the periodic boundary conditions as the stress modulus tends to zero. We show that under suitable conditions on the data the whole sequence of strong probabilistic solutions $(u^{\alpha})$ of the stochastic second grade fluid converges to the unique strong probabilistic solution of the stochastic Navier-Stokes equations.