# Controllability of three-dimensional Navier–Stokes equations and applications

Séminaire Équations aux dérivées partielles (2005-2006)

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topShirikyan, Armen. "Controllability of three-dimensional Navier–Stokes equations and applications." Séminaire Équations aux dérivées partielles (2005-2006): 1-7. <http://eudml.org/doc/11141>.

@article{Shirikyan2005-2006,

abstract = {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.},

author = {Shirikyan, Armen},

journal = {Séminaire Équations aux dérivées partielles},

keywords = {Approximate controllability; exact controllability in projections; 3D Navier–Stokes system; Agrachev–Sarychev method; stationary solutions; irreducibility; complex Ginzburg-Landau equation; ergodicity; external random force; stationary measure},

language = {eng},

pages = {1-7},

publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {Controllability of three-dimensional Navier–Stokes equations and applications},

url = {http://eudml.org/doc/11141},

year = {2005-2006},

}

TY - JOUR

AU - Shirikyan, Armen

TI - Controllability of three-dimensional Navier–Stokes equations and applications

JO - Séminaire Équations aux dérivées partielles

PY - 2005-2006

PB - Centre de mathématiques Laurent Schwartz, École polytechnique

SP - 1

EP - 7

AB - 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.

LA - eng

KW - Approximate controllability; exact controllability in projections; 3D Navier–Stokes system; Agrachev–Sarychev method; stationary solutions; irreducibility; complex Ginzburg-Landau equation; ergodicity; external random force; stationary measure

UR - http://eudml.org/doc/11141

ER -

## References

top- A. Agrachev, S. Kuksin, A. Sarychev, and A. Shirikyan, On finite-dimensional projections of distributions for solutions of randomly forced PDE’s, Preprint (2006). Zbl1177.60059
- 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. Zbl1075.93014MR2127744
- A. A. Agrachev and A. V. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys. (2006), to appear. Zbl1105.93008MR2231685
- F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Probab. Theory Related Fields 102 (1995), no. 3, 367–391. Zbl0831.60072MR1339739
- A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Commun. Math. Phys. (2006), to appear. Zbl1105.93016MR2231968
- A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire (2006), to appear. Zbl1119.93021MR2334990
- A. Shirikyan, Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations, in preparation (2006). Zbl1221.35287MR2345334
- R. Temam, Navier–Stokes Equations, North-Holland, Amsterdam, 1979. Zbl0383.35057MR603444
- M. I. Vishik and A. V. Fursikov, Mathematical Problems in Statistical Hydromechanics, Kluwer, Dordrecht, 1988. Zbl0688.35077

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