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 qualitative properties of admissible weak solutions for randomly forced NS equations.

Keywords: Approximate controllability, exact controllability in projections, 3D Navier–Stokes system, Agrachev–Sarychev method, stationary solutions, irreducibility.

@article{SEDP_2005-2006____A6_0, author = {Shirikyan, Armen}, title = {Controllability of three-dimensional Navier--Stokes equations and applications}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2005-2006}, note = {talk:6}, language = {en}, url = {http://www.numdam.org/item/SEDP_2005-2006____A6_0} }

Shirikyan, Armen. Controllability of three-dimensional Navier–Stokes equations and applications. Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 6, 7 p. http://www.numdam.org/item/SEDP_2005-2006____A6_0/

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