On the convective Brinkman-Forchheimer equations

Abstract

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: $${\partial }_t\mathbf{u}-\hat{I}¼\hat{I}”\mathbf{u}+(\mathbf{u}\cdot \nabla )\mathbf{u}+\hat{I}\pm \mathbf{u}+{\hat{I}}^2|\mathbf{u}{|}^{r-1}\mathbf{u}+\nabla p=\mathbf{f},\nabla \cdot \mathbf{u}=0,$$ (for all values of ${\hat{I}}^2>0$ and $\hat{I}¼>0$, whenever the absorption exponent $r>3$ and $2{\hat{I}}^2\hat{I}¼\ge 1$, 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.

arXiv admin note: text overlap with arXiv:2008.08577