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

Abstract

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:

\begin{document}$ \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*} $\end{document}

in bounded domains \begin{document}$ \Omega\subset\mathbb{R}^d $\end{document} (\begin{document}$ d = 2, 3 $\end{document}) with smooth boundary, where \begin{document}$ \alpha, \beta, \mu>0 $\end{document} and \begin{document}$ r\in[1, \infty) $\end{document}. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function \begin{document}$ \boldsymbol{u} $\end{document}, the pressure gradient \begin{document}$ \nabla p $\end{document} and the vector-valued function \begin{document}$ \boldsymbol{f} $\end{document}. We prove the well-posedness result (existence, uniqueness and stability) of an inverse problem for 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for \begin{document}$ r\geq 1 $\end{document} in two dimensions and for \begin{document}$ r \geq 3 $\end{document} 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.