Controllability of 3D incompressible Euler equations by a finite-dimensional external force
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 677-694.

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.

DOI: 10.1051/cocv/2009017
Classification: 35Q35, 93C20
Keywords: controllability, 3D incompressible Euler equations, Agrachev-Sarychev method
@article{COCV_2010__16_3_677_0,
     author = {Nersisyan, Hayk},
     title = {Controllability of {3D} incompressible {Euler} equations by a finite-dimensional external force},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {677--694},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {3},
     year = {2010},
     doi = {10.1051/cocv/2009017},
     mrnumber = {2674632},
     zbl = {1193.35141},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2009017/}
}
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
PY  - 2010
SP  - 677
EP  - 694
VL  - 16
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2009017/
DO  - 10.1051/cocv/2009017
LA  - en
ID  - COCV_2010__16_3_677_0
ER  - 
%0 Journal Article
%A Nersisyan, Hayk
%T Controllability of 3D incompressible Euler equations by a finite-dimensional external force
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2010
%P 677-694
%V 16
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2009017/
%R 10.1051/cocv/2009017
%G en
%F COCV_2010__16_3_677_0
Nersisyan, Hayk. Controllability of 3D incompressible Euler equations by a finite-dimensional external force. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 677-694. doi : 10.1051/cocv/2009017. http://archive.numdam.org/articles/10.1051/cocv/2009017/

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

[2] A. Agrachev and A. Sarychev, Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing. Comm. Math. Phys. 265 (2006) 673-697. | Zbl

[3] 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. | Zbl

[4] P. Constantin and C. Foias, Navier-Stokes Equations. University of Chicago Press, Chicago, USA (1988). | Zbl

[5] J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155-188. | Zbl

[6] D.E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge, UK (1996). | Zbl

[7] 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.

[8] A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys 54 (1999) 93-146. | Zbl

[9] O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1-44. | Numdam | Zbl

[10] G. Lorentz, Approximation of Functions. Chelsea Publishing Co., New York, USA (1986). | Zbl

[11] S.S. Rodrigues, Navier-Stokes equation on the rectangle: controllability by means of low mode forcing. J. Dyn. Control Syst. 12 (2006) 517-562. | Zbl

[12] A. Shirikyan, Approximate controllability of three-dimensional Navier-Stokes equations. Comm. Math. Phys. 266 (2006) 123-151. | Zbl

[13] A. Shirikyan, Exact controllability in projections for three-dimensional Navier-Stokes equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire 24 (2007) 521-537. | Numdam | Zbl

[14] A. Shirikyan, Euler equations are not exactly controllable by a finite-dimensional external force. Physica D 237 (2008) 1317-1323. | Zbl

[15] M.E. Taylor, Partial Differential Equations, III. Springer-Verlag, New York (1996). | Zbl

[16] R. Temam, Local existence of C solution of the Euler equation of incompressible perfect fluids. Lect. Notes Math. 565 (1976) 184-194. | Zbl

Cited by Sources: