On the convective Brinkman-Forchheimer equations

The convective Brinkman–Forchheimer equations or the Navier–Stokes equations with damping in bounded or periodic domains $\subset {\mathbb{R}}^d$, $2\le d\le 4$ are considered in this work. The existence and uniqueness of a global weak solution in the Leray-Hopf sense satisfying the energy equality to the system: $${\mathrm{\partial }}_t\mathit{u}-\mu \mathrm{\Delta }\mathit{u}+(\mathit{u}\cdot \mathrm{\nabla })\mathit{u}+\alpha \mathit{u}+\beta \mathrm{\mid }\mathit{u}{\mathrm{\mid }}^{r-1}\mathit{u}+\mathrm{\nabla }p=\mathit{f},\mathrm{\nabla }\cdot \mathit{u}=0,$$ (for all values of $\beta >0$ and $\mu >0$, whenever the absorption exponent $r>3$ and $2\beta \mu \ge 1$, for the critical case $r=3$) is proved. We exploit the monotonicity as well as the demicontinuity properties of the linear and nonlinear operators and the Minty-Browder technique in the proofs. Finally, we discuss the existence of global-in-time strong solutions to such systems in periodic domains.