Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity

Abstract

In this work, we carry out the asymptotic analysis of the two dimensional convective Brinkman-Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and Poincar\'e domains (using asymptotic compactness property). In Poincar\'e domains, for $r=1,2$ and $3$, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors for the 2D CBF equations. We consider an expanding sequence of simply connected, bounded and smooth subdomains $\Omega_m$ of the Poincar\'e domain $\Omega$ such that $\Omega_m\to\Omega$ as $m\to\infty$. If $\mathscr{A}_m$ and $\mathscr{A}$ are the global attractors of the 2D CBF equations corresponding to $\Omega$ and $\Omega_m$, respectively, then we show that for large enough $m$, the global attractor $\mathscr{A}_m$ enters into any neighborhood $\mathcal{U}(\mathscr{A})$ of $\mathscr{A}$. The presence of Darcy term in the CBF equations helps us to obtain the above mentioned results in general unbounded domains also. Finally, we discuss about the quasi-stability property of the semigroup associated with the 2D CBF equations in bounded domains and establish the existence of finite fractal dimensional global as well as exponential attractors for $r\in[1,\infty)$.