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.
Mots clés : 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{\textendash}Stokes} equations and applications}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"}, note = {talk:6}, pages = {1--7}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2005-2006}, language = {en}, url = {http://www.numdam.org/item/SEDP_2005-2006____A6_0/} }
TY - JOUR AU - Shirikyan, Armen TI - Controllability of three-dimensional Navier–Stokes equations and applications JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:6 PY - 2005-2006 SP - 1 EP - 7 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2005-2006____A6_0/ LA - en ID - SEDP_2005-2006____A6_0 ER -
%0 Journal Article %A Shirikyan, Armen %T Controllability of three-dimensional Navier–Stokes equations and applications %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:6 %D 2005-2006 %P 1-7 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_2005-2006____A6_0/ %G en %F SEDP_2005-2006____A6_0
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://www.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