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

Abstract

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 and acting volume force are known. However, prior knowledge of and is often impossible in practical engineering applications. This leads to the question of whether the divergence‐free convection field and acting volume force 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 and , 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 abstract regularization framework for the considered inverse problem and establish the existence of a solution.