Existence and uniqueness of time-periodic solutions of the 2D and 3D convective Brinkman-Forchheimer extended Darcy equations

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 $\mathrm{\Omega }\subset {\mathbb{R}}^d$, $d\in \{2,3\}$, \begin{align*}\frac{\partial\boldsymbol{v}}{\partial t}-μΔ\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+α\boldsymbol{v}+β\vert \boldsymbol{v}\vert^{r-1}\boldsymbol{v}+γ\vert \boldsymbol{v}\vert ^{q-1}\boldsymbol{v}+\nabla p=\boldsymbol{g},\ \nabla\cdot\boldsymbol{v}=0, \end{align*} where $\mu ,\alpha ,\beta >0$, $\gamma \in \mathbb{R}$, $r,q\in [1,\mathrm{\infty })$ with $r>q\ge 1$ and $\mathit{g}$ is an external forcing term. For $r\ge 1$, under periodic forcing, we establish the \emph{existence of time-periodic global weak solutions} to the system by employing \emph{Faedo-Galerkin approximations}, together with the \emph{Banach-Alaoglu theorem}, the \emph{Aubin-Lions-Simon compactness lemma}, and the \emph{Lions-Magenes lemma}. The \emph{existence of periodic solutions} for the Faedo-Galerkin approximated problem is obtained via \emph{Brouwer’s fixed point theorem}. In the \emph{supercritical} case $(r>3)$ and the \emph{critical} case ($r=3$), we prove the \emph{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 \emph{uniqueness of strong solutions} typically requires \emph{smallness assumptions} on the external force.