Controllability of three-dimensional Navier–Stokes equations and applications

Armen Shirikyan

Séminaire Équations aux dérivées partielles (2005-2006)

  • page 1-7

Abstract

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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.

How to cite

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Shirikyan, 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

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  1. 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
  2. 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
  3. 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
  4. 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
  5. A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Commun. Math. Phys. (2006), to appear. Zbl1105.93016MR2231968
  6. A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire (2006), to appear. Zbl1119.93021MR2334990
  7. A. Shirikyan, Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations, in preparation (2006). Zbl1221.35287MR2345334
  8. R. Temam, Navier–Stokes Equations, North-Holland, Amsterdam, 1979. Zbl0383.35057MR603444
  9. M. I. Vishik and A. V. Fursikov, Mathematical Problems in Statistical Hydromechanics, Kluwer, Dordrecht, 1988. Zbl0688.35077

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