# Controllability of 3D incompressible Euler equations by a finite-dimensional external force

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 3, page 677-694
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topNersisyan, Hayk. "Controllability of 3D incompressible Euler equations by a finite-dimensional external force." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 677-694. <http://eudml.org/doc/250730>.

@article{Nersisyan2010,

abstract = {
In this paper, we study the
control system associated with the incompressible 3D Euler system.
We show that the velocity field and pressure of the fluid are
exactly controllable in projections by the same finite-dimensional
control. Moreover, the velocity is approximately controllable.
We also prove that 3D Euler
system is not exactly controllable by a finite-dimensional
external force.
},

author = {Nersisyan, Hayk},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Controllability; 3D incompressible Euler equations; Agrachev-Sarychev method; controllability},

language = {eng},

month = {7},

number = {3},

pages = {677-694},

publisher = {EDP Sciences},

title = {Controllability of 3D incompressible Euler equations by a finite-dimensional external force},

url = {http://eudml.org/doc/250730},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Nersisyan, Hayk

TI - Controllability of 3D incompressible Euler equations by a finite-dimensional external force

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/7//

PB - EDP Sciences

VL - 16

IS - 3

SP - 677

EP - 694

AB -
In this paper, we study the
control system associated with the incompressible 3D Euler system.
We show that the velocity field and pressure of the fluid are
exactly controllable in projections by the same finite-dimensional
control. Moreover, the velocity is approximately controllable.
We also prove that 3D Euler
system is not exactly controllable by a finite-dimensional
external force.

LA - eng

KW - Controllability; 3D incompressible Euler equations; Agrachev-Sarychev method; controllability

UR - http://eudml.org/doc/250730

ER -

## References

top- A. Agrachev and A. Sarychev, Navier–Stokes equations controllability by means of low modes forcing. J. Math. Fluid Mech.7 (2005) 108–152. Zbl1075.93014
- A. Agrachev and A. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing. Comm. Math. Phys.265 (2006) 673–697. Zbl1105.93008
- J.T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys.94 (1984) 61–66. Zbl0573.76029
- P. Constantin and C. Foias, Navier–Stokes Equations. University of Chicago Press, Chicago, USA (1988). Zbl0687.35071
- J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl.75 (1996) 155–188. Zbl0848.76013
- D.E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge, UK (1996). Zbl0865.46020
- E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov and J.P. Puel, Local exact controllability of the Navier–Stokes system. J. Math. Pures Appl.83 (2004) 1501–1542. Zbl1267.93020
- A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier–Stokes and Boussinesq equations. Russian Math. Surveys54 (1999) 93–146.
- O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV5 (2000) 1–44. Zbl0940.93012
- G. Lorentz, Approximation of Functions. Chelsea Publishing Co., New York, USA (1986). Zbl0643.41001
- S.S. Rodrigues, Navier–Stokes equation on the rectangle: controllability by means of low mode forcing. J. Dyn. Control Syst.12 (2006) 517–562. Zbl1105.35085
- A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations. Comm. Math. Phys.266 (2006) 123–151. Zbl1105.93016
- A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire24 (2007) 521–537. Zbl1119.93021
- A. Shirikyan, Euler equations are not exactly controllable by a finite-dimensional external force. Physica D237 (2008) 1317–1323. Zbl1143.76393
- M.E. Taylor, Partial Differential Equations, III. Springer-Verlag, New York (1996).
- R. Temam, Local existence of ${C}^{\infty}$ solution of the Euler equation of incompressible perfect fluids. Lect. Notes Math.565 (1976) 184–194.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.