In this work, we investigate the existence and uniqueness of solutions to the following 2D and 3D convective Brinkman–Forchheimer extended Darcy equations defined on a bounded smooth domain \(\Omega \subset \mathbb {R}^d\) , \(d\in \{2,3\}\) , \(\begin{aligned} \frac{\partial \varvec{v}}{\partial t}-\mu \Delta \varvec{v}+(\varvec{v}\cdot \nabla )\varvec{v}+\alpha \varvec{v}+\beta \vert \varvec{v}\vert ^{r-1}\varvec{v}+\gamma \vert \varvec{v}\vert ^{q-1}\varvec{v}+\nabla p=\varvec{g},\ \nabla \cdot \varvec{v}=0, \end{aligned}\) where \(\mu ,\alpha ,\beta >0\) , \(\gamma \in \mathbb {R}\) , \(r,q\in [1,\infty )\) with \(r>q\ge 1\) and \(\varvec{g}\) is an external forcing term. For \(r \ge 1 \) , under periodic forcing, we establish the existence of time-periodic global weak solutions to the system by employing Faedo–Galerkin approximations, together with the Banach–Alaoglu theorem, the Aubin–Lions–Simon compactness lemma, and the Lions–Magenes lemma. The existence of periodic solutions for the Faedo–Galerkin approximated problem is obtained via Brouwer’s fixed point theorem. In the super-critical case \(( r > 3 )\) and the critical case ( \( r = 3\) ), we prove the uniqueness of the global weak solution without imposing any smallness condition on the external forcing. This constitutes a new result compared to the classical 2D Navier–Stokes equations with periodic inputs, for which the uniqueness of strong solutions typically requires smallness assumptions on the external force.
