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.

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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     title = {Controllability of three-dimensional {Navier{\textendash}Stokes} equations and applications},
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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. http://archive.numdam.org/item/SEDP_2005-2006____A6_0/

[AKSS06] A. Agrachev, S. Kuksin, A. Sarychev, and A. Shirikyan, On finite-dimensional projections of distributions for solutions of randomly forced PDE’s, Preprint (2006). | MR

[AS05] A. A. Agrachev and A. V. Sarychev, Navier–Stokes equations: controllability by means of low modes forcing, J. Math. Fluid Mech. 7 (2005), 108–152. | MR | Zbl

[AS06] A. A. Agrachev and A. V. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys. (2006), to appear. | MR | Zbl

[FG95] F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Probab. Theory Related Fields 102 (1995), no. 3, 367–391. | MR | Zbl

[Shi06a] A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Commun. Math. Phys. (2006), to appear. | MR | Zbl

[Shi06b] A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire (2006), to appear. | EuDML | Numdam | MR | Zbl

[Shi06c] A. Shirikyan, Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations, in preparation (2006). | MR

[Tem79] R. Temam, Navier–Stokes Equations, North-Holland, Amsterdam, 1979. | MR | Zbl

[VF88] M. I. Vishik and A. V. Fursikov, Mathematical Problems in Statistical Hydromechanics, Kluwer, Dordrecht, 1988. | MR | Zbl