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Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in periodic domains: $$\frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ where $r\geq1$. We prove that the global attractor of the above system is a singleton under small forcing intensity ($r\geq 1$ for $n=2$ and $r\geq 3$ for $n=3$ with $2βμ\geq 1$ for $r=n=3$). After perturbing the above system with additive or multiplicative white noise, the random attractor does not have a singleton structure. But we obtain that the random attractor for 2D stochastic CBF equations with additive and multiplicative white noise converges towards the deterministic singleton attractor for $1\leq r\leq 2$ and $1\leq r<\infty$, respectively, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). Interestingly in the case of 3D stochastic CBF equations perturbed by multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for $3\leq r\leq 5$ ($2βμ\geq 1$ for $r=3$), when the coefficient of random perturbation converges to zero.