{"title":"Well-posedness of an inverse problem for two- and three-dimensional convective Brinkman-Forchheimer equations with the final overdetermination","authors":"Pardeep Kumar, M. T. Mohan","doi":"10.3934/ipi.2022024","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> \\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*} $\\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in bounded domains <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\Omega\\subset\\mathbb{R}^d $\\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id=\"M2\">\\begin{document}$ d = 2, 3 $\\end{document}</tex-math></inline-formula>) with smooth boundary, where <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\alpha, \\beta, \\mu>0 $\\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M4\">\\begin{document}$ r\\in[1, \\infty) $\\end{document}</tex-math></inline-formula>. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\boldsymbol{u} $\\end{document}</tex-math></inline-formula>, the pressure gradient <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\nabla p $\\end{document}</tex-math></inline-formula> and the vector-valued function <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\boldsymbol{f} $\\end{document}</tex-math></inline-formula>. We prove the well-posedness result (existence, uniqueness and stability) of an inverse problem for 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for <inline-formula><tex-math id=\"M8\">\\begin{document}$ r\\geq 1 $\\end{document}</tex-math></inline-formula> in two dimensions and for <inline-formula><tex-math id=\"M9\">\\begin{document}$ r \\geq 3 $\\end{document}</tex-math></inline-formula> in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inverse Problems and Imaging","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/ipi.2022024","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 3
Abstract
In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:
in bounded domains \begin{document}$ \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.
期刊介绍:
Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing.
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