Approximations of stochastic Navier–Stokes equations

Abstract

In this paper we show that solutions of two-dimensional stochastic Navier-Stokes equations driven by Brownian motion can be approximated by stochastic Navier-Stokes equations forced by pure jump noise/random kicks.

Approximations of stochastic 3D tamed Navier-Stokes equations

“…To prove the above main result, we adopt the method from [25]. We firstly show the result holds under the following stronger condition Hypothesis H5, then we relax it to Hypothesis H4.…”

“…However, their results can not cover some important models in fluid mechanics such as stochastic 2D Navier-Stokes equations, 3D tamed Navier-Stokes equations and so on. For the stochastic 2D Navier-Stokes equations, Shang and Zhang [25] established such an approximation on the state space D([0, T ]; H). In this paper, we aim to prove the same result for the stochastic 3D tamed Navier-Stokes equations on the state space D([0, T ]; H 1 ).…”

“…In this paper, we aim to prove the same result for the stochastic 3D tamed Navier-Stokes equations on the state space D([0, T ]; H 1 ). To achieve it, we proceeds along the same lines as [25]. However, the treatment of nonlinear terms is different from that in [25], as the cancellation property does not hold in H 1 .…”

“…To achieve it, we proceeds along the same lines as [25]. However, the treatment of nonlinear terms is different from that in [25], as the cancellation property does not hold in H 1 . Moreover, as an important part of the proof, we need to show the tightness of the approximating equations on the space D([0, T ]; H 1 ), which requires some estimates of high order Sobolev norms, such as • H 1 , • H 2 .…”