Global Solvability of Inverse Problem for Linear Kelvin-Voigt Equations With Memory

Abstract

In this paper, the inverse problem for a linear system of Kelvin-Voigt equations with memory describing the dynamics of a viscoelastic incompressible non-newtonian fluid is considered. In the inverse problem under consideration, along with the solution (velocity and fluid pressure) of the equation, it is also required to find the unknown (intensity of the external force) on the right side, which depends only on the time variable. Definitions of weak and strong solutions are given. Weak and strong solutions of the set inverse problems satisfy the boundary condition of sliding at the boundary. The sliding boundary condition gives a mathematical and physical character to the study of a linear system of Kelvin-Voigt equations with memory. The applicability of the Faedo-Galerkin method for this type of system of equations is analyzed. With the help of the Faedo-Galerkin method, the global theorem of the existence of solutions to the presented inverse problem is proved in a weak and strong generalized sense. To prove the theorem of the existence of a solution "as a whole"in time, it is associated with obtaining a priori estimates, the constants in which depend only on the data of the problem and the magnitude of the time interval. And also the uniqueness theorem of the solutions of the considered inverse problems for a linear system of Kelvin-Voigt equations with memory is obtained and proved.