On the inverse problem of parameter identification in the steady Oseen model with unilateral and frictional‐type boundary conditions

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 104; no. 12
• Main Authors: Cen, Jinxia, Khan, Akhtar A., Yao, Jen‐Chih, Zeng, Shengda
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.12.2024
• Subjects: Boundary conditions; Existence theorems; Fluid flow; Free convection; Incompressible flow; Incompressible fluids; Inverse problems; Optimization; Parameter identification; Regularization
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.202300427

Recently, Migórski‐Dudek  investigated a steady Oseen flow for a generalized Newtonian incompressible fluid with unilateral and frictional‐type boundary conditions. They established an existence theorem for the steady Oseen model when the divergence‐free convection field b$\mathbf {b}$ and acting volume force f$\mathbf {f}$ are known. However, prior knowledge of b$\mathbf {b}$ and f$\mathbf {f}$ is often impossible in practical engineering applications. This leads to the question of whether the divergence‐free convection field b$\mathbf {b}$ and acting volume force f$\mathbf {f}$ can be determined. The main objective of this paper is to provide a positive response to this challenging and intriguing question. Specifically, we aim to formulate an inverse problem for identifying b$\mathbf {b}$ and f$\mathbf {f}$, characterized by a nonlinear and nonsmooth regularized optimization problem. Initially, we introduce a variational selection mapping for the steady Oseen model. We then establish the boundedness and generalized continuity of this variational selection. Finally, by leveraging optimization theory, convex analysis, and nonsmooth analysis, we develop an regularization framework for the considered inverse problem and establish the existence of a solution.