Determination of the right-hand side of the Navier–Stokes system of equations and inverse problems for the thermal convection equations

Inverse problem for an evolution equation with a quadratic nonlinearity in the Hilbert space is considered. The problem is, given the values of certain functionals of the solution, to find at each point in time the right-hand side that is a linear combination of those functionals. Sufficient conditions for the nonlocal (in time) existence of a solution (on the whole time interval) are established. An application to the inverse problems for the three-dimensional thermal convection equations of viscous incompressible fluid is considered. Unique nonlocal (in terms of time) solvability of the problem of determining the density of heat sources under the regularity condition of the initial data and sufficiently large dimension of the observation space is proved.