Approximate controllability of three-dimensional Navier–Stokes equations

We formulate two results on controllability properties of the 3D Navier– Stokes (NS) system. They concern the approximate controllability and exact controllability in finite-dimensional projections of the problem in question. As a consequence, we obtain the existence of a strong solution of the Cauchy problem for the 3D NS system with an arbitrary initial function and a large class of right-hand sides. We also discuss some qual-itative properties of admissible weak solutions for randomly forced NS equations