American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/0020-7225(74)90042-1.  Google Scholar

[2]

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

[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.  doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[4]

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

[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.  doi: 10.1016/S0252-9602(13)60051-X.  Google Scholar

[6]

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

[7]

J.-Y. Chemin, Perfect Incompressible Fluids,, Clarendon Press, (1998).   Google Scholar

[8]

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

[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.  doi: 10.3934/dcds.2009.23.765.  Google Scholar

[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.  doi: 10.1002/mma.1383.  Google Scholar

[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.  doi: 10.1016/j.na.2010.06.029.  Google Scholar

[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.  doi: 10.1016/j.jde.2010.03.016.  Google Scholar

[13]

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

[14]

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

[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.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[16]

H. Inoue, K. Matsuura and M. Ŏtani, Strong solutions of magneto-micropolar fluid equation,, in Discrete and continuous dynamical systems, (2002), 439.   Google Scholar

[17]

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

[18]

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

[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.  doi: 10.2307/2152750.  Google Scholar

[20]

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

[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.  doi: 10.1155/S1085337599000287.  Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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).  doi: 10.1186/1687-2770-2012-139.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

K. Yamazaki, On the global regularity of N-dimensional generalized Boussinesq system,, Appl. Math., ().   Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[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.  doi: 10.1002/mma.967.  Google Scholar

show all references

References:
[1]

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

[2]

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

[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.  doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[4]

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

[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.  doi: 10.1016/S0252-9602(13)60051-X.  Google Scholar

[6]

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

[7]

J.-Y. Chemin, Perfect Incompressible Fluids,, Clarendon Press, (1998).   Google Scholar

[8]

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

[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.  doi: 10.3934/dcds.2009.23.765.  Google Scholar

[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.  doi: 10.1002/mma.1383.  Google Scholar

[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.  doi: 10.1016/j.na.2010.06.029.  Google Scholar

[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.  doi: 10.1016/j.jde.2010.03.016.  Google Scholar

[13]

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

[14]

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

[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.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[16]

H. Inoue, K. Matsuura and M. Ŏtani, Strong solutions of magneto-micropolar fluid equation,, in Discrete and continuous dynamical systems, (2002), 439.   Google Scholar

[17]

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

[18]

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

[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.  doi: 10.2307/2152750.  Google Scholar

[20]

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

[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.  doi: 10.1155/S1085337599000287.  Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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).  doi: 10.1186/1687-2770-2012-139.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

K. Yamazaki, On the global regularity of N-dimensional generalized Boussinesq system,, Appl. Math., ().   Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[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.  doi: 10.1002/mma.967.  Google Scholar

[1]

Jinbo Geng, Xiaochun Chen, Sadek Gala. On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583
[2]

Hiroshi Inoue, Kei Matsuura, Mitsuharu Ôtani. Strong solutions of magneto-micropolar fluid equation. Conference Publications, 2003, 2003 (Special) : 439-448. doi: 10.3934/proc.2003.2003.439
[3]

Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048
[4]

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
[5]

Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090
[6]

Bo-Qing Dong, Jiahong Wu, Xiaojing Xu, Zhuan Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4133-4162. doi: 10.3934/dcds.2018180
[7]

Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062
[8]

Bo-Qing Dong, Zhi-Min Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 765-784. doi: 10.3934/dcds.2009.23.765
[9]

Yong Zhou, Jishan Fan. Regularity criteria of strong solutions to a problem of magneto-elastic interactions. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1697-1704. doi: 10.3934/cpaa.2010.9.1697
[10]

Tomoyuki Suzuki. Regularity criteria in weak spaces in terms of the pressure to the MHD equations. Conference Publications, 2011, 2011 (Special) : 1335-1343. doi: 10.3934/proc.2011.2011.1335
[11]

Francesca Bucci, Irena Lasiecka. Regularity of boundary traces for a fluid-solid interaction model. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 505-521. doi: 10.3934/dcdss.2011.4.505
[12]

Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585
[13]

Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078
[14]

Jishan Fan, Kun Zhao. Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3949-3967. doi: 10.3934/dcdsb.2018119
[15]

Hugo Beirão da Veiga. Turbulence models, $p-$fluid flows, and $W^{2, L}$ regularity of solutions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 769-783. doi: 10.3934/cpaa.2009.8.769
[16]

Luis A. Caffarelli, Alexis F. Vasseur. The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 409-427. doi: 10.3934/dcdss.2010.3.409
[17]

Jingrui Su. Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3423-3434. doi: 10.3934/dcds.2017145
[18]

Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127
[19]

Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164
[20]

Yong Zhou, Jishan Fan. Regularity criteria for a magnetohydrodynamic-$\alpha$ model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 309-326. doi: 10.3934/cpaa.2011.10.309

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences