Controllability of three-dimensional Navier–Stokes equations and applications
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 6, 7 p.

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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.

Classification : 35Q30, 60H15, 76D05, 93B05, 93C20
Mots-clés : Approximate controllability, exact controllability in projections, 3D Navier–Stokes system, Agrachev–Sarychev method, stationary solutions, irreducibility.
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     author = {Shirikyan, Armen},
     title = {Controllability of three-dimensional {Navier{\textendash}Stokes} equations and applications},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
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     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2005-2006},
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Shirikyan, Armen. Controllability of three-dimensional Navier–Stokes equations and applications. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 6, 7 p. https://geodesic-test.mathdoc.fr/item/SEDP_2005-2006____A6_0/