A Quasi-Explicit Method Applied to Missing Boundary Data Reconstruction for the Stokes System - Abdulaziz H. Alharbi; Fadhel Jday; Abdulrahman B. Albidah; Ali M. Alhartomi (Axioms)
In this paper, we study the data completion problem for the Cauchy–Stokes equation in a cylindrical domain, <semantics>Ω</semantics>. Neumann and Dirichlet boundary conditions are prescribed on part of the overdetermined boundary, <semantics>Γ0</semantics>, and the goal is to complete the data on the other part of the boundary, <semantics>Γa</semantics>. Here, <semantics>Γ0</semantics> and <semantics>Γa</semantics> represent the side faces of the cylinder <semantics>Ω</semantics>. This problem is known to be ill-posed and is formulated as an optimal control problem with a regularized cost function. To directly approximate the missing data on <semantics>Γa</semantics>, we employ the method of factorization of elliptic boundary value problems. This technique allows the factorization of a boundary value problem into a product of parabolic problems. It is successfully applied to the optimality system in this work, yielding new and significant results.