Approximate controllability and Irreducibility of the transition semigroup associated with Convective Brinkman-Forchheimer extended Darcy Equations

Abstract

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}-μΔ\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+α\boldsymbol{y}+β\vert \boldsymbol{y}\vert^{r-1}\boldsymbol{y}+γ\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 $r>q\ge 1$ and $\mathbf{u}$ 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.