Approximations of 2D and 3D Stochastic Convective Brinkman-Forchheimer Extended Darcy Equations

Abstract

In this article, we consider two- and three- dimensional stochastic convective Brinkman-Forchheimer extended Darcy (CBFeD) equations

$$ \frac{\partial \boldsymbol{u}}{\partial t}-\mu \Delta \boldsymbol{u}+( \boldsymbol{u}\cdot \nabla )\boldsymbol{u}+\alpha |\boldsymbol{u}|^{q-1} \boldsymbol{u}+\beta |\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p= \boldsymbol{f},\ \nabla \cdot \boldsymbol{u}=0, $$

on a torus, where \(\mu ,\beta >0\), \(\alpha \in \mathbb{R}\), \(r\in [1,\infty )\) and \(q\in [1,r)\). The goal is to show that the solutions of 2D and 3D stochastic CBFeD equations driven by Brownian motion can be approximated by 2D and 3D stochastic CBFeD equations forced by pure jump noise/random kicks on the state space \(\mathrm{D}([0,T];\mathbb{H})\). For the cases \(d=2\), \(r\in [1,\infty )\) and \(d=3\), \(r\in (3,\infty )\), by using minimal regularity assumptions on the noise coefficient, the results are established for any \(\mu ,\beta >0\). For the case \(d=r=3\), the same results are obtained for \(2\beta \mu \geq 1\).