This paper deals with the distributed and boundary controllability of the so called Leray-α model. This is a regularized variant of the Navier-Stokes system (α is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-α equations are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the controls converge as α → 0+ to a null control of the Navier-Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.
Mots clés : null controllability, Carleman inequalities, Leray-αmodel, Navier−Stokes equations
@article{COCV_2014__20_4_1181_0,
author = {Araruna, F\'agner D. and Fern\'andez-Cara, Enrique and Souza, Diego A.},
title = {Uniform local null control of the {Leray-}$\alpha $ model},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1181--1202},
publisher = {EDP-Sciences},
volume = {20},
number = {4},
year = {2014},
doi = {10.1051/cocv/2014011},
zbl = {1297.93031},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2014011/}
}
TY - JOUR AU - Araruna, Fágner D. AU - Fernández-Cara, Enrique AU - Souza, Diego A. TI - Uniform local null control of the Leray-$\alpha $ model JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 1181 EP - 1202 VL - 20 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014011/ DO - 10.1051/cocv/2014011 LA - en ID - COCV_2014__20_4_1181_0 ER -
%0 Journal Article %A Araruna, Fágner D. %A Fernández-Cara, Enrique %A Souza, Diego A. %T Uniform local null control of the Leray-$\alpha $ model %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 1181-1202 %V 20 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014011/ %R 10.1051/cocv/2014011 %G en %F COCV_2014__20_4_1181_0
Araruna, Fágner D.; Fernández-Cara, Enrique; Souza, Diego A. Uniform local null control of the Leray-$\alpha $ model. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1181-1202. doi : 10.1051/cocv/2014011. http://www.numdam.org/articles/10.1051/cocv/2014011/
[1] , and , Optimal control. Contemporary Soviet Mathematics. Translated from the Russian by V.M. Volosov. Consultants Bureau, New York (1987). | MR | Zbl
[2] , and , On the control of the Burgers-alpha model. Adv. Differ. Eq. 18 (2013) 935-954. | MR | Zbl
[3] and , Local null controllability of the N-dimensional Navier−Stokes system with N − 1 scalar controls in an arbitrary control domain. J. Math. Fluid Mech. 15 (2013) 139-153. | MR | Zbl
[4] , , and , On a Leray-α model of turbulence. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 461 (2005) 629-649. | MR | Zbl
[5] and , Navier−Stokes equations, Chicago Lect. Math. University of Chicago Press, Chicago, IL (1988). | MR | Zbl
[6] , On the controllability of the 2-D incompressible Navier−Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1995/96) 35-75. | Numdam | MR | Zbl
[7] and , Global exact controllability of the 2D Navier−Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429-448. | MR | Zbl
[8] and , Null controllability of the N-dimensional Stokes system with N − 1 scalar controls. J. Differ. Eq. 246 (2009) 2908-2921. | MR | Zbl
[9] and , Local null controllability of the three-dimensional navier−stokes system with a distributed control having two vanishing components. Preprint (2012). | MR
[10] and , Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 1, Collection du Commissariat à l'Énergie Atomique: Série Scientifique. [Collection of the Atomic Energy Commission: Science Series]. Masson, Paris (1984). | MR | Zbl
[11] , , and , Local exact controllability for the one-dimensional compressible Navier−Stokes equation. Arch. Ration. Mech. Anal. 206 (2012) 189-238. | MR
[12] and , Null controllability of the Burgers system with distributed controls. Systems Control Lett. 56 (2007) 366-372. | MR | Zbl
[13] , , and , Local exact controllability of the Navier−Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | Zbl
[14] , , and , Some controllability results for the N-dimensional Navier−Stokes and Boussinesq systems with N − 1 scalar controls. SIAM J. Control Optim. 45 (2006) 146-173. | MR | Zbl
[15] and , On the Navier−Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964) 269-315. | MR | Zbl
[16] and , On fractional powers of the Stokes operator. Proc. Japan Acad. 46 (1970) 1141-1143. | MR | Zbl
[17] and , Exact controllability of the Navier−Stokes and Boussinesq equations. Uspekhi Mat. Nauk 54 (1999) 93-146. | MR | Zbl
[18] and , Controllability of evolution equations, vol. 34 of Lect. Notes Ser. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996). | MR | Zbl
[19] and , Estimates for the LANS-α, Leray-α and Bardina models in terms of a Navier−Stokes Reynolds number. Indiana Univ. Math. J. 57 (2008) 2761-2773. | MR | Zbl
[20] and , On the uniform controllability of the Burgers equation. SIAM J. Control Optim. 46 (2007) 1211-1238. | MR | Zbl
[21] , and , Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Commun. Pure Appl. Anal. 8 (2009) 311-333. | MR | Zbl
[22] , Local exact controllability to the trajectories of the Boussinesq system. Ann. Inst. Henri Poincaré Anal. Non Linéaire 23 (2006) 29-61. | Numdam | MR | Zbl
[23] and , Remarks on global controllability for the Burgers equation with two control forces. Annal. Inst. Henri Poincaré Anal. Non Linéaire 24 (2007) 897-906. | Numdam | MR | Zbl
[24] , and , A result concerning the global approximate controllability of the Navier−Stokes system in dimension 3. J. Math. Pures Appl. 98 (2012) 689-709. | MR | Zbl
[25] , Remarks on exact controllability for the Navier−Stokes equations. ESAIM: COCV 6 (2001) 39-72. | Numdam | MR | Zbl
[26] , Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934) 193-248. | JFM | MR
[27] , Remarques sur la controlâbilite approchée, in Spanish-French Conference on Distributed-Systems Control, Spanish. Univ. Málaga, Málaga (1990) 77-87. | MR | Zbl
[28] , Compact sets in the space Lp(0,T;B). Annal. Mat. Pura Appl. 146 (1987) 65-96. | MR | Zbl
[29] , An introduction to Sobolev spaces and interpolation spaces, vol. 3 of Lect. Notes of the Unione Matematica Italiana. Springer, Berlin (2007). | MR | Zbl
[30] , Navier−Stokes equations. Theory and numerical analysis. Vol. 2 of Studies Math. Appl. North-Holland Publishing Co., Amsterdam (1977). | Zbl
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