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May  2015, 35(5): 2193-2207. doi: 10.3934/dcds.2015.35.2193

Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity

Kazuo Yamazaki 1,

1. 

Department of Mathematics, Washington State University, Pullman, WA 99164-3113

Received  January 2014 Revised  October 2014 Published  December 2014

We study the two-dimensional magneto-micropolar fluid system. Making use of the structure of the system, we show that with zero angular viscosity the solution triple remains smooth for all time.
Keywords: Magneto-micropolar fluid, micropolar fluid, regularity criteria, global regularity, Besov spaces..
Mathematics Subject Classification: Primary: 35B65, 35Q35; Secondary: 35Q8.
Citation: Kazuo Yamazaki. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2193-2207. doi: 10.3934/dcds.2015.35.2193
References:
[1]

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

[2]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.

[3]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[4]

C. Cao, J. Wu and B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588-602. doi: 10.1137/130937718.

[5]

M. Chen, Global well-posedness of the 2D incompressible micropolar fluid flows with partial viscosity and angular viscosity, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 929-935. doi: 10.1016/S0252-9602(13)60051-X.

[6]

Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724. doi: 10.1016/j.jde.2011.09.035.

[7]

J.-Y. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998.

[8]

B.-Q. Dong and Z.-M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 50 (2009), 103525, 13pp. doi: 10.1063/1.3245862.

[9]

B.-Q. Dong and Z.-M. Chen, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete Contin. Dyn. Syst., 23 (2009), 765-784. doi: 10.3934/dcds.2009.23.765.

[10]

B.Q. Dong, Y. Jia and Z.-M. Chen, Pressure regularity criteria of the three-dimensional micropolar fluid flows, Math. Methods Appl. Sci., 34 (2011), 595-606. doi: 10.1002/mma.1383.

[11]

B.-Q. Dong and W. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal., 73 (2010), 2334-2341. doi: 10.1016/j.na.2010.06.029.

[12]

B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213. doi: 10.1016/j.jde.2010.03.016.

[13]

A. C. Eringen, Simple microfluids, Int. Engng. Sci., 2 (1964), 205-217. doi: 10.1016/0020-7225(64)90005-9.

[14]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18. doi: 10.1512/iumj.1967.16.16001.

[15]

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

[16]

H. Inoue, K. Matsuura and M. Ŏtani, Strong solutions of magneto-micropolar fluid equation, in Discrete and continuous dynamical systems, 2003; Dynamical systems and differential equations, Wilmington, NC (2002), 439-448.

[17]

Q. Jiu and J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys., (2014), 1-11. doi: 10.1007/s00033-014-0415-8.

[18]

G. Lukaszewicz, On nonstationary flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 12 (1988), 83-97.

[19]

G. Lukaszewicz, On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 13 (1989), 105-120. doi: 10.2307/2152750.

[20]

G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.

[21]

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.

[22]

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

[23]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[24]

C. V. Tran, X. Yu and Z. Zhai, On global regularity of 2D generalized magnetohydrodynamics equations, J. Differential Equations, 254 (2013), 4194-4216. doi: 10.1016/j.jde.2013.02.016.

[25]

Y. Wang, Regularity criterion for a weak solution to the three-dimensional magneto-micropolar fluid equations, Bound. Value Probl., 2013 (2013), 12pp. doi: 10.1186/1687-2770-2013-58.

[26]

J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.

[27]

J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.

[28]

Z. Xiang and H. Yang, On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative, Bound. Value Probl., 139 (2012), 14pp. doi: 10.1186/1687-2770-2012-139.

[29]

L. Xue, Wellposedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34 (2011), 1760-1777. doi: 10.1002/mma.1491.

[30]

N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Math. Meth. Appl. Sci., 28 (2005), 1507-1526. doi: 10.1002/mma.617.

[31]

K. Yamazaki, Remarks on the global regularity of the two-dimensional magnetohydrodynamics system with zero dissipation, Nonliear Anal., 94 (2014), 194-205. doi: 10.1016/j.na.2013.08.020.

[32]

K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity, Appl. Math. Lett., 29 (2014), 46-51. doi: 10.1016/j.aml.2013.10.014.

[33]

K. Yamazaki, On the global regularity of two-dimensional generalized magnetohydrodynamics system, J. Math. Anal. Appl., 416 (2014), 99-111. doi: 10.1016/j.jmaa.2014.02.027.

[34]

K. Yamazaki, On the global regularity of N-dimensional generalized Boussinesq system, Appl. Math., to appear.

[35]

K. Yamazaki, $(N-1)$ velocity components condition for the generalized MHD system in $N$-dimension, Kinet. Relat. Models, 7 (2014), 779-792.

[36]

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.

[37]

B. Yuan, On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space, Proc. Amer. Math. Soc., 138 (2010), 2025-2036. doi: 10.1090/S0002-9939-10-10232-9.

[38]

B. Yuan and L. Bai, Remarks on global regularity of 2D generalized MHD equations, J. Math. Anal. Appl., 413 (2014), 633-640. doi: 10.1016/j.jmaa.2013.12.024.

[39]

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

show all references

References:
[1]

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

[2]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.

[3]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[4]

C. Cao, J. Wu and B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588-602. doi: 10.1137/130937718.

[5]

M. Chen, Global well-posedness of the 2D incompressible micropolar fluid flows with partial viscosity and angular viscosity, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 929-935. doi: 10.1016/S0252-9602(13)60051-X.

[6]

Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724. doi: 10.1016/j.jde.2011.09.035.

[7]

J.-Y. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998.

[8]

B.-Q. Dong and Z.-M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 50 (2009), 103525, 13pp. doi: 10.1063/1.3245862.

[9]

B.-Q. Dong and Z.-M. Chen, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete Contin. Dyn. Syst., 23 (2009), 765-784. doi: 10.3934/dcds.2009.23.765.

[10]

B.Q. Dong, Y. Jia and Z.-M. Chen, Pressure regularity criteria of the three-dimensional micropolar fluid flows, Math. Methods Appl. Sci., 34 (2011), 595-606. doi: 10.1002/mma.1383.

[11]

B.-Q. Dong and W. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal., 73 (2010), 2334-2341. doi: 10.1016/j.na.2010.06.029.

[12]

B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213. doi: 10.1016/j.jde.2010.03.016.

[13]

A. C. Eringen, Simple microfluids, Int. Engng. Sci., 2 (1964), 205-217. doi: 10.1016/0020-7225(64)90005-9.

[14]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18. doi: 10.1512/iumj.1967.16.16001.

[15]

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

[16]

H. Inoue, K. Matsuura and M. Ŏtani, Strong solutions of magneto-micropolar fluid equation, in Discrete and continuous dynamical systems, 2003; Dynamical systems and differential equations, Wilmington, NC (2002), 439-448.

[17]

Q. Jiu and J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys., (2014), 1-11. doi: 10.1007/s00033-014-0415-8.

[18]

G. Lukaszewicz, On nonstationary flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 12 (1988), 83-97.

[19]

G. Lukaszewicz, On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 13 (1989), 105-120. doi: 10.2307/2152750.

[20]

G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.

[21]

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.

[22]

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

[23]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[24]

C. V. Tran, X. Yu and Z. Zhai, On global regularity of 2D generalized magnetohydrodynamics equations, J. Differential Equations, 254 (2013), 4194-4216. doi: 10.1016/j.jde.2013.02.016.

[25]

Y. Wang, Regularity criterion for a weak solution to the three-dimensional magneto-micropolar fluid equations, Bound. Value Probl., 2013 (2013), 12pp. doi: 10.1186/1687-2770-2013-58.

[26]

J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.

[27]

J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.

[28]

Z. Xiang and H. Yang, On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative, Bound. Value Probl., 139 (2012), 14pp. doi: 10.1186/1687-2770-2012-139.

[29]

L. Xue, Wellposedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34 (2011), 1760-1777. doi: 10.1002/mma.1491.

[30]

N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Math. Meth. Appl. Sci., 28 (2005), 1507-1526. doi: 10.1002/mma.617.

[31]

K. Yamazaki, Remarks on the global regularity of the two-dimensional magnetohydrodynamics system with zero dissipation, Nonliear Anal., 94 (2014), 194-205. doi: 10.1016/j.na.2013.08.020.

[32]

K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity, Appl. Math. Lett., 29 (2014), 46-51. doi: 10.1016/j.aml.2013.10.014.

[33]

K. Yamazaki, On the global regularity of two-dimensional generalized magnetohydrodynamics system, J. Math. Anal. Appl., 416 (2014), 99-111. doi: 10.1016/j.jmaa.2014.02.027.

[34]

K. Yamazaki, On the global regularity of N-dimensional generalized Boussinesq system, Appl. Math., to appear.

[35]

K. Yamazaki, $(N-1)$ velocity components condition for the generalized MHD system in $N$-dimension, Kinet. Relat. Models, 7 (2014), 779-792.

[36]

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.

[37]

B. Yuan, On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space, Proc. Amer. Math. Soc., 138 (2010), 2025-2036. doi: 10.1090/S0002-9939-10-10232-9.

[38]

B. Yuan and L. Bai, Remarks on global regularity of 2D generalized MHD equations, J. Math. Anal. Appl., 413 (2014), 633-640. doi: 10.1016/j.jmaa.2013.12.024.

[39]

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

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