This paper deals with the unique solvability of some inverse problems for nonlinear Navier-Stokes-Voigt (Kelvin-Voigt) system with memory that governs the flow of incompressible viscoelastic non-Newtonian fluids. The inverse problems that study here, consist of determining a time dependent intensity of the density of external forces, along with a velocity and a pressure of fluids. As an additional information, two types of integral overdetermination conditions over space domain are considered. The system supplemented also with an initial and one of the boundary conditions: stick and slip boundary conditions. For all inverse problems, under suitable assumptions on the data, the global and local in time existence and uniqueness of weak and strong solutions were established.

In this paper, we study two inverse problems for the nonlinear Boussinesq system for incompressible viscoelastic non-isothermal Kelvin-Voigt fluids. The studying inverse problems consist of determining an intensities of density of external forces and heat source under given integral overdetermination conditions. Two types of boundary conditions for the velocity v are considered: sticking and sliding conditions on boundary. In both cases of these boundary conditions, the local and global in time existence and uniqueness of weak and strong solutions are established under suitable assumptions on the data. The large time behavior of weak solutions is also studied.