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Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity
1. | Department of Mathematics, Washington State University, Pullman, WA 99164-3113 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
J.-Y. Chemin, Perfect Incompressible Fluids,, Clarendon Press, (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).
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.
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.
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.
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.
doi: 10.1016/j.jde.2010.03.016. |
[13] |
A. C. Eringen, Simple microfluids,, Int. Engng. Sci., 2 (1964), 205.
doi: 10.1016/0020-7225(64)90005-9. |
[14] |
A. C. Eringen, Theory of micropolar fluids,, J. Math. Mech., 16 (1966), 1.
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.
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, (2002), 439.
|
[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. |
[18] |
G. Lukaszewicz, On nonstationary flows of asymmetric fluids,, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 12 (1988), 83.
|
[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. |
[20] |
G. Lukaszewicz, Micropolar Fluids, Theory and Applications,, Birkhäuser, (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.
doi: 10.1155/S1085337599000287. |
[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. |
[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. |
[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. |
[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. |
[26] |
J. Wu, The generalized MHD equations,, J. Differential Equations, 195 (2003), 284.
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.
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).
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.
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.
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.
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.
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.
doi: 10.1016/j.jmaa.2014.02.027. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
J.-Y. Chemin, Perfect Incompressible Fluids,, Clarendon Press, (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).
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.
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.
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.
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.
doi: 10.1016/j.jde.2010.03.016. |
[13] |
A. C. Eringen, Simple microfluids,, Int. Engng. Sci., 2 (1964), 205.
doi: 10.1016/0020-7225(64)90005-9. |
[14] |
A. C. Eringen, Theory of micropolar fluids,, J. Math. Mech., 16 (1966), 1.
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.
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, (2002), 439.
|
[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. |
[18] |
G. Lukaszewicz, On nonstationary flows of asymmetric fluids,, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 12 (1988), 83.
|
[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. |
[20] |
G. Lukaszewicz, Micropolar Fluids, Theory and Applications,, Birkhäuser, (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.
doi: 10.1155/S1085337599000287. |
[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. |
[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. |
[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. |
[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. |
[26] |
J. Wu, The generalized MHD equations,, J. Differential Equations, 195 (2003), 284.
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.
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).
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.
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.
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.
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.
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.
doi: 10.1016/j.jmaa.2014.02.027. |
[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. |
[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. |
[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. |
[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. |
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