×

Exact controllability of Galerkin’s approximations of micropolar fluids. (English) Zbl 1186.35153

Summary: We consider the nonlinear model describing micropolar fluid in a bounded smooth region of \( \mathbb{R}^{N} (N = 2,3)\) with distributed controls supported in small subset of this domain. Under suitable assumptions on the Galerkin basis, we introduce Galerkin’s approximations for the controllable micropolar fluid system. By using the Hilbert Uniqueness Method in combination with a fixed point argument, we prove the exact controllability result for this finite-dimensional system.

MSC:

35Q35 PDEs in connection with fluid mechanics
93B05 Controllability
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. Ariman and M. Turk, On steady and pulsatile flow of blood, J. Appl. Mech., 41 (1974), 1-7. · Zbl 0356.76080
[2] J. L. Boldrini, B. Climent-Ezquerra, M. A. Rojas-Medar and M. D. Rojas-Medar, On an Iterative Method for Approximate Solutions of a Generalized Boussinesq Model, to appear in Journal Mathematical Fluid Mechanics. · Zbl 1270.76076
[3] Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications].
[4] C. Calmelet-Eluhu and D. R. Majumdar, Flow of a micropolar fluid through a circular cylinder subject to longitudinal and torsional oscillations, Math. Comput. Modelling 27 (1998), no. 8, 69 – 78. · Zbl 1076.76508 · doi:10.1016/S0895-7177(98)00044-2
[5] Duane W. Condiff and John S. Dahler, Fluid mechanical aspects of antisymmetric stress, Phys. Fluids 7 (1964), 842 – 854. · Zbl 0125.15801 · doi:10.1063/1.1711295
[6] Jean-Michel Coron, On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Contrôle Optim. Calc. Var. 1 (1995/96), 35 – 75. · Zbl 0872.93040
[7] Jean-Michel Coron and Andrei V. Fursikov, Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary, Russian J. Math. Phys. 4 (1996), no. 4, 429 – 448. · Zbl 0938.93030
[8] D. Dupuy, G. P. Panasenko, and R. Stavre, Asymptotic methods for micropolar fluids in a tube structure, Math. Models Methods Appl. Sci. 14 (2004), no. 5, 735 – 758. · Zbl 1076.76004 · doi:10.1142/S0218202504003428
[9] A. Cemal Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966), 1 – 18.
[10] A. Cemal Eringen, Simple microfluids, Internat. J. Engrg. Sci. 2 (1964), 205 – 217 (English, with French, German, Italian and Russian summaries). · Zbl 0136.45003 · doi:10.1016/0020-7225(64)90005-9
[11] E. Fernández-Cara and S. Guerrero, Local exact controllability of micropolar fluids, J. Math. Fluid Mech. 9 (2007), no. 3, 419 – 453. · Zbl 1133.35423 · doi:10.1007/s00021-005-0207-1
[12] E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov, and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9) 83 (2004), no. 12, 1501 – 1542 (English, with English and French summaries). · Zbl 1267.93020 · doi:10.1016/j.matpur.2004.02.010
[13] A. V. Fursikov, Optimal control of distributed systems. Theory and applications, Translations of Mathematical Monographs, vol. 187, American Mathematical Society, Providence, RI, 2000. Translated from the 1999 Russian original by Tamara Rozhkovskaya. · Zbl 1027.93500
[14] A. V. Fursikov, M. D. Gunzburger, and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: the three-dimensional case, SIAM J. Control Optim. 43 (2005), no. 6, 2191 – 2232. · Zbl 1076.76029 · doi:10.1137/S0363012904400805
[15] A. V. Fursikov, M. D. Gunzburger, and L. S. Hou, Optimal Dirichlet control and inhomogeneous boundary value problems for the unsteady Navier-Stokes equations, Control and partial differential equations (Marseille-Luminy, 1997) ESAIM Proc., vol. 4, Soc. Math. Appl. Indust., Paris, 1998, pp. 97 – 116. · Zbl 0920.76020 · doi:10.1051/proc:1998023
[16] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. · Zbl 0862.49004
[17] F. Guillén-González, M. A. Rojas-Medar and M. A. Rodrıguez-Bellido, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr., 282 (6) (2009), 846-867. · Zbl 1173.35033
[18] Oleg Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var. 6 (2001), 39 – 72. · Zbl 0961.35104 · doi:10.1051/cocv:2001103
[19] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. · Zbl 0184.52603
[20] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 9, Masson, Paris, 1988 (French). Perturbations. [Perturbations]. · Zbl 0653.93003
[21] Jacques-Louis Lions and Enrique Zuazua, Contrôlabilité exacte des approximations de Galerkin des équations de Navier-Stokes, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 9, 1015 – 1021 (French, with English and French summaries). · Zbl 0894.93020 · doi:10.1016/S0764-4442(97)87878-0
[22] Jacques-Louis Lions and Enrique Zuazua, Exact boundary controllability of Galerkin’s approximations of Navier-Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 4, 605 – 621. · Zbl 1053.93009
[23] Jacques-Louis Lions and Enrique Zuazua, On the cost of controlling unstable systems: the case of boundary controls, J. Anal. Math. 73 (1997), 225 – 249. · Zbl 0892.93036 · doi:10.1007/BF02788145
[24] Grzegorz Łukaszewicz, Micropolar fluids, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. Theory and applications. · Zbl 0923.76003
[25] E. Ortega-Torres, M. A. Rojas-Medar, On the regularity for solutions of the micropolar fluid equations, to appear in Rend. Sem. Mat. Univ. Padova. · Zbl 1372.35246
[26] L. Petrosyan, Some Problems of Fluid Mechanics with Antisymmetric Stress Tensor, Erevan, 1984 (in Russian).
[27] A. S. Popel, S. A. Regirer, P. I. Usick, A continuum model of blood flow, Biorheology, 11 (1974), 427-437.
[28] Marko A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution, Math. Nachr. 188 (1997), 301 – 319. · Zbl 0893.76006 · doi:10.1002/mana.19971880116
[29] R. Stavre, The control of the pressure for a micropolar fluid, Z. Angew. Math. Phys. 53 (2002), no. 6, 912 – 922. Dedicated to Eugen Soós. · Zbl 1036.76012 · doi:10.1007/PL00012619
[30] Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. · Zbl 0426.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.