Inverse problems for nonlinear Navier–Stokes–Voigt system with memory.

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. • The inverse problems are equivalent to the direct problems for a nonlinear parabolic equation with nonlinear nonlocal operator of the function u. • The inverse problems have unique weak and strong solutions in local time. • The following inverse problems have unique weak and strong solutions in global time in particular case. [ABSTRACT FROM AUTHOR]