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 \begin{equation*} \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, \end{equation*} 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 on the state space $\mathrm{D}([0,T];\mathbb{H})$. The results are established for $d=2,r\in[1,\infty)$ and $d=3,r\in[3,\infty)$ with $2\beta\mu\geq 1$ for $d=r=3,$ and by using less regular assumptions on the noise coefficient.