Uniform local null control of the Leray-$\alpha$ model

Araruna, Fágner D.; Fernández-Cara, Enrique; Souza, Diego A.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1181-1202.

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.

[1] 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). | MR | Zbl

[2] F.D. Araruna, E. Fernández-Cara and D.A. Souza, On the control of the Burgers-alpha model. Adv. Differ. Eq. 18 (2013) 935-954. | MR | Zbl

[3] N. Carreño and S. Guerrero, 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] A. Cheskidov, D.D. Holm, E. Olson and E.S. Titi, On a Leray-α model of turbulence. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 461 (2005) 629-649. | MR | Zbl

[5] P. Constantin and C. Foias, Navier−Stokes equations, Chicago Lect. Math. University of Chicago Press, Chicago, IL (1988). | MR | Zbl

[6] J.-M. Coron, 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] J.-M. Coron and A.V. Fursikov, 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] J.-M. Coron and S. Guerrero, Null controllability of the N-dimensional Stokes system with N − 1 scalar controls. J. Differ. Eq. 246 (2009) 2908-2921. | MR | Zbl

[9] 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). | MR

[10] 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). | MR | Zbl

[11] S. Ervedoza, O. Glass, S. Guerrero and J.-P. Puel, Local exact controllability for the one-dimensional compressible Navier−Stokes equation. Arch. Ration. Mech. Anal. 206 (2012) 189-238. | MR

[12] E. Fernández-Cara and S. Guerrero, Null controllability of the Burgers system with distributed controls. Systems Control Lett. 56 (2007) 366-372. | MR | Zbl

[13] E. Fernández-Cara, S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier−Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | Zbl

[14] E. Fernández-Cara, S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, 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] H. Fujita and T. Kato, On the Navier−Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964) 269-315. | MR | Zbl

[16] H. Fujita and H. Morimoto, On fractional powers of the Stokes operator. Proc. Japan Acad. 46 (1970) 1141-1143. | MR | Zbl

[17] A.V. Fursikov and O.Y. Imanuvilov, Exact controllability of the Navier−Stokes and Boussinesq equations. Uspekhi Mat. Nauk 54 (1999) 93-146. | MR | Zbl

[18] 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). | MR | Zbl

[19] J.D. Gibbon and D.D. Holm, 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] O. Glass and S. Guerrero, On the uniform controllability of the Burgers equation. SIAM J. Control Optim. 46 (2007) 1211-1238. | MR | Zbl

[21] M. González-Burgos, S. Guerrero and J.-P. Puel, 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] S. Guerrero, 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] S. Guerrero and O.Y. Imanuvilov, 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] S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, 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] O.Y. Imanuvilov, Remarks on exact controllability for the Navier−Stokes equations. ESAIM: COCV 6 (2001) 39-72. | Numdam | MR | Zbl

[26] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934) 193-248. | JFM | MR

[27] J.-L. Lions, Remarques sur la controlâbilite approchée, in Spanish-French Conference on Distributed-Systems Control, Spanish. Univ. Málaga, Málaga (1990) 77-87. | MR | Zbl

[28] J. Simon, Compact sets in the space Lp(0,T;B). Annal. Mat. Pura Appl. 146 (1987) 65-96. | MR | Zbl

[29] L. Tartar, An introduction to Sobolev spaces and interpolation spaces, vol. 3 of Lect. Notes of the Unione Matematica Italiana. Springer, Berlin (2007). | MR | Zbl

[30] R. Temam, Navier−Stokes equations. Theory and numerical analysis. Vol. 2 of Studies Math. Appl. North-Holland Publishing Co., Amsterdam (1977). | Zbl