Global well-posedness of a three-dimensional Brinkman-Forchheimer-Bénard convection model in porous media

Abstract

We consider three-dimensional (3D) Boussinesq convection system of an incompressible fluid in a closed sample of a porous medium. Specifically, we introduce and analyze a 3D Brinkman-Forchheimer-Bénard convection problem describing the behavior of an incompressible fluid in a porous medium between two plates heated from the bottom and cooled from the top. We show the existence and uniqueness of global in-time solutions, and the existence of absorbing balls in L 2 and H 1 . Eventually, we comment on the applicability of a data assimilation algorithm to our system.

An analytical and computational study of the incompressible Toner-Tu Equations

“…However, there is a significant literature on a more general class of equations where the u|u| 2 -term is replaced by u|u| 2δ , which is the case in the Brinkman-Forchheimer extended Darcy model arising in porous media. The paper by Titi and Trabelsi [33] contains a wide literature survey; but we also refer the reader to [32,[34][35][36][37][38][39][40][41][42][43][44]. When δ > 1 , the invariant scaling property of the Navier-Stokes equations is broken.…”

“…Given the close relationship between the ITT equations and the incompressible Navier-Stokes equations, a formal approach is taken on the understanding that the standard Leray-Hopf weaksolution machinery, derived for the Navier-Stokes equations, is already in place [32,33]. In this notation the energy H 0 and and enstrophy H 1 are:…”

“…An exception is the work of Zanger, Löwen, and Saal on the regularity of solutions of the TTSH equations [31]. It turns out that similar PDEs have been studied using the methods of analysis in the context of the NS equations with an absorption term [32] or the Brinkman-Forchheimer-extended Darcy model of porous media [32][33][34][35][36][37][38][39][40][41][42][43][44]. The major difference is that these models have a nonlinearity that breaks the NS-invariance enjoyed by the ITT equations.…”

“…1. We take a formal approach on the understanding that the standard Leray-Hopf weaksolution machinery, derived for the Navier-Stokes equations, is already in place [32,33]. 2.…”

An analytical and computational study of the incompressible Toner–Tu Equations

“…Although these active-matter and active-fluid PDEs have been studied intensively over the past two decades from a physical perspective, with the results of these investigations having undergone wide experimental comparison, detailed methods of Navier-Stokes analysis [26][27][28][29][30][31][32] have not generally been applied directly to the ITT equations. 1 However, models related to the NSEs with an absorption term have been studied [34], including the Brinkman-Forchheimer-extended Darcy model of porous media [34][35][36][37][38][39][40][41][42][43][44][45][46]. The major difference is that these models possess a nonlinearity that breaks the Navier-Stokes invariance enjoyed by the ITT equations.…”