Existence, uniqueness and stability of an inverse problem for two-dimensional convective Brinkman–Forchheimer equations with the integral overdetermination

In this article, we study an inverse problem for the following convective Brinkman–Forchheimer (CBF) equations: ut-μΔu+(u·∇)u+αu+β|u|r-1u+∇p=F:=fg,∇·u=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{u}_t-\mu \Delta \varvec{u}+(\varvec{u}\cdot \nabla )\varvec{u}+\alpha \varvec{u} +\beta |\varvec{u}|^{r-1}\varvec{u}+\nabla p=\varvec{F}:=f \varvec{g}, \ \ \ \nabla \cdot \varvec{u}=0, \end{aligned}$$\end{document}in a bounded domain Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^2$$\end{document} with smooth boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega$$\end{document}, where α,β,μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta ,\mu >0$$\end{document} and r∈[1,3]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in [1,3]$$\end{document}. The investigated inverse problem consists of reconstructing the vector-valued velocity function u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{u}$$\end{document}, the pressure field p and the scalar function f. For the divergence free initial data u0∈L2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{u}_0 \in \mathbb {L}^2(\Omega )$$\end{document}, we prove the existence of a solution to the inverse problem for two-dimensional CBF equations with the integral overdetermination condition, by showing the existence of a unique fixed point for an equivalent operator equation (using an extension of the contraction mapping theorem). Moreover, we establish the uniqueness and Lipschitz stability results of the solution to the inverse problem for 2D CBF equations with r∈[1,3]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \in [1,3]$$\end{document}.