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Uniform local null control of the Leray-α model∗∗∗

Published online by Cambridge University Press:  08 August 2014

Fágner D. Araruna ,
Enrique Fernández-Cara  and
Diego A. Souza
Fágner D. Araruna
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa PB, Brasil. [email protected]
Enrique Fernández-Cara
Affiliation:
Departamento EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; [email protected]; [email protected]
Diego A. Souza
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa PB, Brasil. [email protected] Departamento EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; [email protected]; [email protected]
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Abstract

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.

Keywords

Null controllabilityCarleman inequalitiesLeray-αmodelNavier−Stokes equations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1181 - 1202
Copyright
© EDP Sciences, SMAI, 2014

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References

V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin, Optimal control. Contemporary Soviet Mathematics. Translated from the Russian by V.M. Volosov. Consultants Bureau, New York (1987). Google Scholar
Araruna, F.D., Fernández-Cara, E. and Souza, D.A., On the control of the Burgers-alpha model. Adv. Differ. Eq. 18 (2013) 935–954. Google Scholar
Carreño, N. and Guerrero, S., 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. Google Scholar
Cheskidov, A., Holm, D.D., Olson, E. and Titi, E.S., On a Leray-α model of turbulence. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 461 (2005) 629–649. Google Scholar
P. Constantin and C. Foias, Navier−Stokes equations, Chicago Lect. Math. University of Chicago Press, Chicago, IL (1988). Google Scholar
Coron, J.-M., On the controllability of the 2-D incompressible Navier−Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1995/96) 35–75. Google Scholar
Coron, J.-M. and Fursikov, A.V., Global exact controllability of the 2D Navier−Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429–448. Google Scholar
Coron, J.-M. and Guerrero, S., Null controllability of the N-dimensional Stokes system with N − 1 scalar controls. J. Differ. Eq. 246 (2009) 2908–2921. Google Scholar
J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional navier−stokes system with a distributed control having two vanishing components. Preprint (2012). Google Scholar
R. Dautray and J.-L. Lions, 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). Google Scholar
Ervedoza, S., Glass, O., Guerrero, S. and Puel, J.-P., Local exact controllability for the one-dimensional compressible Navier−Stokes equation. Arch. Ration. Mech. Anal. 206 (2012) 189–238. Google Scholar
Fernández-Cara, E. and Guerrero, S., Null controllability of the Burgers system with distributed controls. Systems Control Lett. 56 (2007) 366–372. Google Scholar
Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Y. and Puel, J.-P., Local exact controllability of the Navier−Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. Google Scholar
Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Y. and Puel, J.-P., 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. Google Scholar
Fujita, H. and Kato, T., On the Navier−Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964) 269–315. Google Scholar
Fujita, H. and Morimoto, H., On fractional powers of the Stokes operator. Proc. Japan Acad. 46 (1970) 1141–1143. Google Scholar
Fursikov, A.V. and Imanuvilov, O.Y., Exact controllability of the Navier−Stokes and Boussinesq equations. Uspekhi Mat. Nauk 54 (1999) 93–146. Google Scholar
A.V. Fursikov and O.Y. Imanuvilov, Controllability of evolution equations, vol. 34 of Lect. Notes Ser. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996). Google Scholar
Gibbon, J.D. and Holm, D.D., 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. Google Scholar
Glass, O. and Guerrero, S., On the uniform controllability of the Burgers equation. SIAM J. Control Optim. 46 (2007) 1211–1238. Google Scholar
González-Burgos, M., Guerrero, S. and Puel, J.-P., 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. Google Scholar
Guerrero, S., Local exact controllability to the trajectories of the Boussinesq system. Ann. Inst. Henri Poincaré Anal. Non Linéaire 23 (2006) 29–61. Google Scholar
Guerrero, S. and Imanuvilov, O.Y., Remarks on global controllability for the Burgers equation with two control forces. Annal. Inst. Henri Poincaré Anal. Non Linéaire 24 (2007) 897–906. Google Scholar
Guerrero, S., Imanuvilov, O.Y. and Puel, J.-P., A result concerning the global approximate controllability of the Navier−Stokes system in dimension 3. J. Math. Pures Appl. 98 (2012) 689–709. Google Scholar
Imanuvilov, O.Y., Remarks on exact controllability for the Navier−Stokes equations. ESAIM: COCV 6 (2001) 39–72. Google Scholar
Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 (1934) 193–248. Google Scholar
J.-L. Lions, Remarques sur la controlâbilite approchée, in Spanish-French Conference on Distributed-Systems Control, Spanish. Univ. Málaga, Málaga (1990) 77–87. Google Scholar
Simon, J., Compact sets in the space L p(0,T;B). Annal. Mat. Pura Appl. 146 (1987) 65–96. Google Scholar
L. Tartar, An introduction to Sobolev spaces and interpolation spaces, vol. 3 of Lect. Notes of the Unione Matematica Italiana. Springer, Berlin (2007). Google Scholar
R. Temam, Navier−Stokes equations. Theory and numerical analysis. Vol. 2 of Studies Math. Appl. North-Holland Publishing Co., Amsterdam (1977). Google Scholar