Inverse Problems for Heat Convection System for Incompressible Viscoelastic Fluids

The paper deals the study some inverse source problems for heat convection system which consists of Kelvin–Voigt equations governing an incompressible viscoelastic non-Newtonian flows and a heat equation. The studying inverse problems consist of recovering a time depended intensity \(f(t)\) of a density of external forces and an intensity \(j(t)\) of a heat source, in addition to a velocity \(\mathbf{v}\) , a pressure \(\pi\) , and a temperature \(\theta\) . As an additional information two types of integral overdetermination conditions over the domain are considered. For nonlinear inverse problem, under suitable conditions on the data, the local in time existence and uniqueness of weak and strong solutions are established. Some special cases of original inverse problem also investigated which allow global unique solvability.