Nonlocal problems for one class of nonlinear operator equations that arise in the theory of Sobolev type equations (Q1261271)

A nonlinear operator equation generalizing the incompressible Navier- Stokes equation with an extra inner damping term is studied. It is shown that if certain a priori estimates are known then there exists a unique globally defined solution of the initial value or periodic or almost periodic problem, respectively. The abstract equation in question can also be obtained from a certain class of dissipative equations of Sobolev type.