Conditions for the Existence of a Global Strong Solution to a Class of Nonlinear Evolution Equations in a Hilbert Space

We study a nonlinear operator differential equation in a Hilbert space. This equation represents an abstract model for the system of Navier–Stokes equations. The main result consists in proving the existence of a strong solution to this equation under the condition that a certain other system of equations (related to the original equation) has only the zero solution.