Well-posedness of an inverse problem for two and three dimensional convective Brinkman-Forchheimer equations with the final overdetermination

TL;DR

This paper proves the well-posedness of an inverse problem for convective Brinkman-Forchheimer equations in 2D and 3D, enabling the reconstruction of flow velocity, pressure, and forcing functions under certain conditions.

Contribution

It establishes existence, uniqueness, and stability results for the inverse problem with final overdetermination, extending previous work to nonlinear regimes with fast-growing nonlinearities.

Findings

Proved well-posedness for 2D and 3D cases.

Applied Schauder's fixed point theorem for arbitrary smooth initial data.

Extended results to nonlinearities with $r eq 1$ in 2D and $r eq 3$ in 3D.

Abstract

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations: \begin{align*} \boldsymbol{u}_t-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{F}:=\boldsymbol{f} g, \ \ \ \nabla\cdot\boldsymbol{u}=0, \end{align*} in bounded domains $\Omega\subset\mathbb{R}^d$ ($d=2,3$) with smooth boundary, where $\alpha,\beta,\mu>0$ and $r\in[1,\infty)$. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function $\boldsymbol{u}$, the pressure field $p$ and the vector-valued function $\boldsymbol{f}$. We have proved the well-posedness result (existence, uniqueness and stablility) for the inverse problem for the 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for $r\geq 1$ in two dimensions and for $r \geq 3$ in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities.