Exact controllability in projections for three-dimensional Navier-Stokes equations

The paper is devoted to studying controllability properties for 3D Navier-Stokes equations in a bounded domain. We establish a Sufficient condition under which the problem in question is exactly controllable in any finite-dimensional projection. Out- sufficient condition is verified for any torus in R-3. The proofs are based on a development of a general approach introduced by Agrachev and Sarychev in the 2D case. As a simple consequence of the result on controllability, we show that the Cauchy problem for the 3D Navier-Stokes system has a unique strong solution for any initial function and a large class of external forces. (C) 2006 Elsevier Masson SAS. All rights reserved.