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September  2017, 6(3): 357-379. doi: 10.3934/eect.2017019

Local exact controllability to trajectories of the magneto-micropolar fluid equations

1. 

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

2. 

Faculty of Computer Science and Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam

* Corresponding author: anhctmath@hnue.edu.vn

Dedicated to Prof. Nguyen Manh Hung on the occasion of his 60th birthday

Received  January 2017 Revised  March 2017 Published  July 2017

In this paper we prove the exact controllability to trajectories of the magneto-micropolar fluid equations with distributed controls. We first establish new Carleman inequalities for the associated linearized system which lead to its null controllability. Then, combining the null controllability of the linearized system with an inverse mapping theorem, we deduce the local exact controllability to trajactories of the nonlinear problem.

Citation: Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations & Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019
References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663. doi: 10.1016/0020-7225(74)90042-1. Google Scholar

[2]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Translated from the Russian by V. M. Volosov. Contemporary Soviet Mathematics. Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1. Google Scholar

[3]

M. Badra, Local controllability to trajectories of the magnetohydrodynamic equations, J. Math. Fluid Mech., 16 (2014), 631-660. doi: 10.1007/s00021-014-0186-1. Google Scholar

[4]

V. BarbuT. HavărneanuC. Popa and S. S. Sritharan, Exact controllability for the magnetohydrodynamic equations, Comm. Pure Appl. Math., 56 (2003), 732-783. doi: 10.1002/cpa.10072. Google Scholar

[5]

V. BarbuT. HavărneanuC. Popa and S. S. Sritharan, Local exact controllability for the magnetohydrodynamic equations revisited, Adv. Differential Equations, 10 (2005), 481-504. Google Scholar

[6]

P. Braz e SilvaL. Friz and M. A. Rojas-Medar, Exponential stability for magneto-micropolar fluids, Nonlinear Anal., 143 (2016), 211-223. doi: 10.1016/j.na.2016.05.015. Google Scholar

[7]

R. Dautray and J. -L. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques, vol. 5. INSTN: Collection Enseignement. [INSTN: Teaching Collection]. Masson, Paris, 1988. Google Scholar

[8]

M. DuránJ. Ferreira and M. A. Rojas-Medar, Reproductive weak solutions of magneto-micropolar fluid equations in exterior domains, Math. Comput. Modelling, 35 (2002), 779-791. doi: 10.1016/S0895-7177(02)00049-3. Google Scholar

[9]

E. Fernández-Cara and S. Guerrero, Local exact controllability of micropolar fluids, J. Math. Fluid Mech., 9 (2007), 419-445. doi: 10.1007/s00021-005-0207-1. Google Scholar

[10]

E. Fernández-CaraS. GuerreroO. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl., 83 (2004), 1501-1542. doi: 10.1016/j.matpur.2004.02.010. Google Scholar

[11]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. iv+163 pp. Google Scholar

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Exact local controllability of two-dimensional NavierStokes equations. (Russian) Mat. Sb. 187 (1996), 103-138; translation in Sb. Math. 187 (1996), 1355-1390. doi: 10.1070/SM1996v187n09ABEH000160. Google Scholar

[13]

S. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space, NoDEA Nonlinear Differential Equations Appl., 10 (2011), 583-592. doi: 10.1007/s00030-009-0047-4. Google Scholar

[14]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108. doi: 10.1016/0020-7225(77)90025-8. Google Scholar

[15]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S. Google Scholar

[16]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61. doi: 10.1016/j.anihpc.2005.01.002. Google Scholar

[17]

S. Guerrero and P. Cornilleau, On the local exact controllability of micropolar fuids with few controls, ESAIM Control Optim. Calc. Var., 23 (2017), 637-662. doi: 10.1051/cocv/2016010. Google Scholar

[18]

T. HavǎrneanuC. Popa and S. S. Sritharan, Exact internal controllability for the magnetohydrodynamic equations in multi-connected domains, Adv. Differential Equations, 11 (2006), 893-929. Google Scholar

[19]

T. HavărneanuC. Popa and S. S. Sritharan, Exact internal controllability for the two-dimensional magnetohydrodynamic equations, SIAM J. Control Optim., 46 (2007), 1802-1830. doi: 10.1137/040611884. Google Scholar

[20]

O. Yu. ImanuvilovJ.-P. Puel and M. Yamamoto, Carleman estimates for second order nonhomogeneous parabolic equations, Chin. Ann. Math. Ser. B, 30 (2009), 333-378. doi: 10.1007/s11401-008-0280-x. Google Scholar

[21]

G. Lukaszewicz and W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., 55 (2004), 247-257. doi: 10.1007/s00033-003-1127-7. Google Scholar

[22]

K. Matsura, Exponential attractors for 2D magneto-micropolar fluid flow in a bounded domain, Discrete Contin. Dyn. Syst., suppl. (2005), 634-641. Google Scholar

[23]

W. G. Melo, The magneto-micropolar equations with periodic boundary conditions: solution properties at potential blow-up times, J. Math. Anal. Appl., 435 (2016), 1194-1209. doi: 10.1016/j.jmaa.2015.11.005. Google Scholar

[24]

P. Orliński, The existence of an exponential attractor in magneto-micropolar fluid flow via the $\ell$-trajectories method, Colloq. Math., 132 (2013), 221-238. doi: 10.4064/cm132-2-5. Google Scholar

[25]

E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125. doi: 10.1155/S1085337599000287. Google Scholar

[26]

J. -P. Puel, Controllability of Navier-Stokes equations, Optimization with PDE Constraints, 379-402, Lect. Notes Comput. Sci. Eng., 101, Springer, Cham, 2014. doi: 10.1007/978-3-319-08025-3_12. Google Scholar

[27]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319. doi: 10.1002/mana.19971880116. Google Scholar

[28]

M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: Existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460. doi: 10.5209/rev_REMA.1998.v11.n2.17276. Google Scholar

[29]

W. Sadowski, Upper bound for the number of degrees of freedom for magneto-micropolar flows and turbulence, Internat. J. Engrg. Sci., 41 (2003), 789-800. doi: 10.1016/S0020-7225(02)00283-5. Google Scholar

[30]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces Lecture Notes of the Unione Matematica Italiana, 3. Springer, Berlin; UMI, Bologna, 2007. Google Scholar

[31]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., vol. 2, North-Holland, Amsterdam-New York-Oxford, 1977. Google Scholar

[32]

Y.-Z. Wang and Y.-X. Wang, Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity, Math. Methods Appl. Sci., 34 (2011), 2125-2135. doi: 10.1002/mma.1510. Google Scholar

[33]

B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 1469-1480. doi: 10.1016/S0252-9602(10)60139-7. Google Scholar

[34]

J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Methods Appl. Sci., 31 (2008), 1113-1130. doi: 10.1002/mma.967. Google Scholar

[35]

H. Zhang, Regularity criteria for the 3D magneto-micropolar equations, Acta Math. Appl. Sin., 37 (2014), 487-496. Google Scholar

show all references

References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663. doi: 10.1016/0020-7225(74)90042-1. Google Scholar

[2]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Translated from the Russian by V. M. Volosov. Contemporary Soviet Mathematics. Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1. Google Scholar

[3]

M. Badra, Local controllability to trajectories of the magnetohydrodynamic equations, J. Math. Fluid Mech., 16 (2014), 631-660. doi: 10.1007/s00021-014-0186-1. Google Scholar

[4]

V. BarbuT. HavărneanuC. Popa and S. S. Sritharan, Exact controllability for the magnetohydrodynamic equations, Comm. Pure Appl. Math., 56 (2003), 732-783. doi: 10.1002/cpa.10072. Google Scholar

[5]

V. BarbuT. HavărneanuC. Popa and S. S. Sritharan, Local exact controllability for the magnetohydrodynamic equations revisited, Adv. Differential Equations, 10 (2005), 481-504. Google Scholar

[6]

P. Braz e SilvaL. Friz and M. A. Rojas-Medar, Exponential stability for magneto-micropolar fluids, Nonlinear Anal., 143 (2016), 211-223. doi: 10.1016/j.na.2016.05.015. Google Scholar

[7]

R. Dautray and J. -L. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques, vol. 5. INSTN: Collection Enseignement. [INSTN: Teaching Collection]. Masson, Paris, 1988. Google Scholar

[8]

M. DuránJ. Ferreira and M. A. Rojas-Medar, Reproductive weak solutions of magneto-micropolar fluid equations in exterior domains, Math. Comput. Modelling, 35 (2002), 779-791. doi: 10.1016/S0895-7177(02)00049-3. Google Scholar

[9]

E. Fernández-Cara and S. Guerrero, Local exact controllability of micropolar fluids, J. Math. Fluid Mech., 9 (2007), 419-445. doi: 10.1007/s00021-005-0207-1. Google Scholar

[10]

E. Fernández-CaraS. GuerreroO. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl., 83 (2004), 1501-1542. doi: 10.1016/j.matpur.2004.02.010. Google Scholar

[11]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. iv+163 pp. Google Scholar

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Exact local controllability of two-dimensional NavierStokes equations. (Russian) Mat. Sb. 187 (1996), 103-138; translation in Sb. Math. 187 (1996), 1355-1390. doi: 10.1070/SM1996v187n09ABEH000160. Google Scholar

[13]

S. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space, NoDEA Nonlinear Differential Equations Appl., 10 (2011), 583-592. doi: 10.1007/s00030-009-0047-4. Google Scholar

[14]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108. doi: 10.1016/0020-7225(77)90025-8. Google Scholar

[15]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S. Google Scholar

[16]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61. doi: 10.1016/j.anihpc.2005.01.002. Google Scholar

[17]

S. Guerrero and P. Cornilleau, On the local exact controllability of micropolar fuids with few controls, ESAIM Control Optim. Calc. Var., 23 (2017), 637-662. doi: 10.1051/cocv/2016010. Google Scholar

[18]

T. HavǎrneanuC. Popa and S. S. Sritharan, Exact internal controllability for the magnetohydrodynamic equations in multi-connected domains, Adv. Differential Equations, 11 (2006), 893-929. Google Scholar

[19]

T. HavărneanuC. Popa and S. S. Sritharan, Exact internal controllability for the two-dimensional magnetohydrodynamic equations, SIAM J. Control Optim., 46 (2007), 1802-1830. doi: 10.1137/040611884. Google Scholar

[20]

O. Yu. ImanuvilovJ.-P. Puel and M. Yamamoto, Carleman estimates for second order nonhomogeneous parabolic equations, Chin. Ann. Math. Ser. B, 30 (2009), 333-378. doi: 10.1007/s11401-008-0280-x. Google Scholar

[21]

G. Lukaszewicz and W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., 55 (2004), 247-257. doi: 10.1007/s00033-003-1127-7. Google Scholar

[22]

K. Matsura, Exponential attractors for 2D magneto-micropolar fluid flow in a bounded domain, Discrete Contin. Dyn. Syst., suppl. (2005), 634-641. Google Scholar

[23]

W. G. Melo, The magneto-micropolar equations with periodic boundary conditions: solution properties at potential blow-up times, J. Math. Anal. Appl., 435 (2016), 1194-1209. doi: 10.1016/j.jmaa.2015.11.005. Google Scholar

[24]

P. Orliński, The existence of an exponential attractor in magneto-micropolar fluid flow via the $\ell$-trajectories method, Colloq. Math., 132 (2013), 221-238. doi: 10.4064/cm132-2-5. Google Scholar

[25]

E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125. doi: 10.1155/S1085337599000287. Google Scholar

[26]

J. -P. Puel, Controllability of Navier-Stokes equations, Optimization with PDE Constraints, 379-402, Lect. Notes Comput. Sci. Eng., 101, Springer, Cham, 2014. doi: 10.1007/978-3-319-08025-3_12. Google Scholar

[27]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319. doi: 10.1002/mana.19971880116. Google Scholar

[28]

M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: Existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460. doi: 10.5209/rev_REMA.1998.v11.n2.17276. Google Scholar

[29]

W. Sadowski, Upper bound for the number of degrees of freedom for magneto-micropolar flows and turbulence, Internat. J. Engrg. Sci., 41 (2003), 789-800. doi: 10.1016/S0020-7225(02)00283-5. Google Scholar

[30]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces Lecture Notes of the Unione Matematica Italiana, 3. Springer, Berlin; UMI, Bologna, 2007. Google Scholar

[31]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., vol. 2, North-Holland, Amsterdam-New York-Oxford, 1977. Google Scholar

[32]

Y.-Z. Wang and Y.-X. Wang, Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity, Math. Methods Appl. Sci., 34 (2011), 2125-2135. doi: 10.1002/mma.1510. Google Scholar

[33]

B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 1469-1480. doi: 10.1016/S0252-9602(10)60139-7. Google Scholar

[34]

J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Methods Appl. Sci., 31 (2008), 1113-1130. doi: 10.1002/mma.967. Google Scholar

[35]

H. Zhang, Regularity criteria for the 3D magneto-micropolar equations, Acta Math. Appl. Sin., 37 (2014), 487-496. Google Scholar

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