Existence and uniqueness of time-periodic solutions of the 2D and 3D convective Brinkman-Forchheimer extended Darcy equations
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2026
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| author | Mohan, Manil T. |
| author_facet | Mohan, Manil T. |
| contents | 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 $Ω\subset\mathbb{R}^d$, $d\in\{2,3\}$, \begin{align*}\frac{\partial\boldsymbol{v}}{\partial t}-μΔ\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+α\boldsymbol{v}+β\vert \boldsymbol{v}\vert^{r-1}\boldsymbol{v}+γ\vert \boldsymbol{v}\vert ^{q-1}\boldsymbol{v}+\nabla p=\boldsymbol{g},\ \nabla\cdot\boldsymbol{v}=0, \end{align*} where $μ,α,β>0$, $γ\in\mathbb{R}$, $r,q\in[1,\infty)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_01917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and uniqueness of time-periodic solutions of the 2D and 3D convective Brinkman-Forchheimer extended Darcy equations Mohan, Manil T. Analysis of PDEs 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 $Ω\subset\mathbb{R}^d$, $d\in\{2,3\}$, \begin{align*}\frac{\partial\boldsymbol{v}}{\partial t}-μΔ\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+α\boldsymbol{v}+β\vert \boldsymbol{v}\vert^{r-1}\boldsymbol{v}+γ\vert \boldsymbol{v}\vert ^{q-1}\boldsymbol{v}+\nabla p=\boldsymbol{g},\ \nabla\cdot\boldsymbol{v}=0, \end{align*} where $μ,α,β>0$, $γ\in\mathbb{R}$, $r,q\in[1,\infty)$ 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. |
| title | Existence and uniqueness of time-periodic solutions of the 2D and 3D convective Brinkman-Forchheimer extended Darcy equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.01917 |