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In this work, we consider the two and three-dimensional stochastic convectiveBrinkman-Forchheimer (2D and 3D SCBF) equations driven by irregular additivewhite noise $$\mathrm{d}\boldsymbol{u}-[\mu\Delta\boldsymbol{u}-(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}-\alpha\boldsymbol{u}-\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}-\nablap]\mathrm{d} t=\boldsymbol{f}\mathrm{d} t+\mathrm{d}\mathrm{W},\\nabla\cdot\boldsymbol{u}=0,$$ for $r\in[1,\infty),$ $\mu,\alpha,\beta>0$ inunbounded domains (like Poincar\'e domains) $\mathcal{O}\subset\mathbb{R}^d$($d=2,3$) where $\mathrm{W}(\cdot)$ is a Hilbert space valued Wiener process onsome given filtered probability space, and discuss the asymptotic behavior ofits solution. For $d=2$ with $r\in[1,\infty)$ and $d=3$ with $r\in[3,\infty)$(for $d=r=3$ with $2\beta\mu\geq 1$), we first prove the existence anduniqueness of a weak solution (in the analytic sense) satisfying the energyequality for SCBF equations driven by an irregular additive white noise inPoincar\'e domains by using a Faedo-Galerkin approximation technique. Since theenergy equality for SCBF equations is not immediate, we construct a sequencewhich converges in Lebesgue and Sobolev spaces simultaneously and it helps usto demonstrate the energy equality. Then, we establish the existence of randomattractors for the stochastic flow generated by the SCBF equations. One of thetechnical difficulties connected with the irregular white noise is overcomewith the help of the corresponding Cameron-Martin space (or Reproducing KernelHilbert space). Finally, we address the existence of a unique invariant measurefor 2D and 3D SCBF equations defined on Poincar\'e domains (bounded orunbounded). Moreover, we provide a remark on the extension of the abovementioned results to general unbounded domains also.

Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains