Inverse problem for Navier-Stokes systems with finite-dimensional overdetermination

We consider an inverse problem for an evolution equation with a quadratic nonlinearity. In this problem, one should find the right-hand side belonging at each time to a finitedimensional subspace on the basis of the given projection of the solution onto that subspace. We prove the time-nonlocal solvability of the inverse problem. Under the condition of additional regularity of the original data and a sufficiently large dimension of the observation subspace, we show that the solution of the inverse problem is unique and more smooth. By way of application, we consider the inverse problem for the three-dimensional Navier-Stokes equations describing the dynamics of a viscous incompressible fluid.