On the convective Brinkman-Forchheimer equations
Sagar Gautam, Manil T. Mohan
TL;DR
This work analyzes the convective Brinkman–Forchheimer system in bounded or periodic domains ($2\le d\le 4$) and proves existence and uniqueness of global Leray–Hopf weak solutions that satisfy the energy balance for $r\ge 3$ (with the critical case $r=3$ requiring $2\beta\mu\ge 1$). The authors employ monotone operator theory, demicontinuity, and Minty–Browder techniques to establish solvability, and they further obtain global strong solutions in periodic domains under suitable regularity, leveraging an abstract BV framework. The paper also develops a comprehensive functional-analytic setup (spaces, operators $A$, $B$, $C_r$) and provides detailed energy-estimate machinery, including mollification-based arguments to justify energy equalities. These results extend the classical NSE theory by incorporating damping and porous-medium terms, with applicability to 2–4D periodic and bounded geometries and potential stochastic extensions.
Abstract
The convective Brinkman--Forchheimer equations or the Navier--Stokes equations with damping in bounded or periodic domains $\subset\mathbb{R}^d$, $2\leq d\leq 4$ 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: $$\partial_t\boldsymbol{u}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ (for all values of $β>0$ and $μ>0$, whenever the absorption exponent $r>3$ and $2βμ\geq 1$, 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.