In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a $d$-dimensional torus $\mathbb{ T}^d$: \begin{ align*} \frac{ \partial\boldsymbol{ y}}{ \partial t}-\mu \Delta\boldsymbol{ y}+(\boldsymbol{ y}\cdot\nabla)\boldsymbol{ y}+\alpha\boldsymbol{ y}+\beta\vert \boldsymbol{ y}\vert^{ r-1}\boldsymbol{ y}+\gamma\vert \boldsymbol{ y}\vert ^{ q-1}\boldsymbol{ y}+\nabla p=\boldsymbol{ g}+\boldsymbol{ u},\ \nabla\cdot\boldsymbol{ y}=0, \end{ align*} where $d\in\{ 2,3\}$, $\mu,\alpha,\beta>0$, $\gamma\in\mathbb{ R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$ and $\boldsymbol{ u}$ is the control. For the super critical ($r>3$) and critical ($r=3$ with $2\beta\mu>1$) cases, we first show the approximate controllability of the above system in the usual energy space (divergence-free $\mathbb{ L}^2(\mathbb{ T}^d)$ space). As an application of the approximate controllability result, we establish the irreducibility of the transition semigroup associated with stochastic CBFeD system perturbed by non-degenerate Gaussian noise in the usual energy space by exploiting the regularity of solutions, smooth approximation of the multi-valued map $\mathrm{ sgn}(\cdot)$ a density argument and monotonicity properties of the linear and nonlinear operators.
Categories: math.PR math.AP math.OC