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

TL;DR

This paper proves the existence and uniqueness of time-periodic solutions for the 2D and 3D convective Brinkman-Forchheimer equations, extending results on fluid flow models with nonlinear damping and external forcing.

Contribution

It establishes the existence of time-periodic weak solutions and proves uniqueness in supercritical and critical cases without smallness conditions, advancing understanding of nonlinear Darcy-type equations.

Findings

Existence of time-periodic weak solutions under periodic forcing.

Uniqueness of solutions in supercritical and critical regimes without smallness assumptions.

Application of Faedo-Galerkin method and fixed point theorems to nonlinear PDEs.

Abstract

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 $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, \begin{align*}\frac{\partial\boldsymbol{v}}{\partial t}-\mu \Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+\alpha\boldsymbol{v}+\beta\vert \boldsymbol{v}\vert^{r-1}\boldsymbol{v}+\gamma\vert \boldsymbol{v}\vert ^{q-1}\boldsymbol{v}+\nabla p=\boldsymbol{g},\ \nabla\cdot\boldsymbol{v}=0, \end{align*} where $\mu,\alpha,\beta>0$, $\gamma\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$ and $\boldsymbol{g}$ is an external forcing term. For $r \geq 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.