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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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[35]

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

show all references

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[35]

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

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