(N-1) velocity components condition for the generalized MHD system in N-dimension

Previous Article
KRM Home
This Issue
Next Article
Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres

December 2014, 7(4): 779-792. doi: 10.3934/krm.2014.7.779

$(N-1)$ velocity components condition for the generalized MHD system in $N-$dimension

Kazuo Yamazaki ^1,

1.	Department of Mathematics, Washington State University, Pullman, WA 99164-3113, United States

Received March 2014 Revised August 2014 Published November 2014

Abstract
Related Papers

We study the magnetohydrodynamics system, generalized via a fractional Laplacian. When the domain is in $N-$dimension, $N$ being three, four or five, we show that the regularity criteria of its solution pair may be reduced to $(N-1)$ many velocity field components with the improved integrability condition in comparison to the result in [29]. Furthermore, we extend this result to the three-dimensional magneto-micropolar fluid system.

Keywords: fractional Laplacians., Navier-Stokes equations, magneto-micropolar fluid system, Magnetohydrodynamics system.

Mathematics Subject Classification: Primary: 35B65, 35Q35; Secondary: 35Q8.

Citation: Kazuo Yamazaki. $(N-1)$ velocity components condition for the generalized MHD system in $N-$dimension. Kinetic and Related Models, 2014, 7 (4) : 779-792. doi: 10.3934/krm.2014.7.779

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]	J. Beale, T. Kato and A. Majda, Remarks on breakdown of smooth solutions for the three-dimensional Euler equations, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.
[3]	S. Benbernou, S. Gala and M. A. Ragusa, On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space, Math. Methods Appl. Sci., 37 (2013), 2320-2325. doi: 10.1002/mma.2981.
[4]	L. C. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc., 130 (2002), 3585-3595. doi: 10.1090/S0002-9939-02-06697-2.
[5]	C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719.
[6]	C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020.
[7]	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.
[8]	A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
[9]	S. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 181-194. doi: 10.1007/s00030-009-0047-4.
[10]	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.
[11]	C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.
[12]	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.
[13]	X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, II, Kinet. Relat. Models, 7 (2014), 291-304. doi: 10.3934/krm.2014.7.291.
[14]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.
[15]	I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012.
[16]	G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.
[17]	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.
[18]	P. Penel and M. Pokorný, On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353. doi: 10.1007/s00021-010-0038-6.
[19]	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.
[20]	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.
[21]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.
[22]	J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.
[23]	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.
[24]	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.
[25]	K. Yamazaki, Regularity criteria of porous media equation in terms of one partial derivative or pressure field, Commun. Math. Sci., to appear.
[26]	K. Yamazaki, Regularity criteria of supercritical beta-generalized quasi-geostrophic equation in terms of partial derivatives, Electron. J. Differential Equations, 2013 (2013), 1-12.
[27]	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.
[28]	K. Yamazaki, Regularity criteria of MHD system involving one velocity component and one current density component, J. Math. Fluid Mech., 16 (2014), 551-570. doi: 10.1007/s00021-014-0178-1.
[29]	K. Yamazaki, Remarks on the regularity criteria of three-dimensional magnetohydrodynamics system in terms of two velocity field components, J. Math. Phys., 55 (2014), 031505, 16pp. doi: 10.1063/1.4868277.
[30]	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.
[31]	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.
[32]	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.
[33]	Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886. doi: 10.3934/dcds.2005.12.881.
[34]	Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbbR^{3}$, Proc. Amer. Math. Soc., 134 (2006), 149-156. doi: 10.1090/S0002-9939-05-08312-7.
[35]	Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014.

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]	J. Beale, T. Kato and A. Majda, Remarks on breakdown of smooth solutions for the three-dimensional Euler equations, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.
[3]	S. Benbernou, S. Gala and M. A. Ragusa, On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space, Math. Methods Appl. Sci., 37 (2013), 2320-2325. doi: 10.1002/mma.2981.
[4]	L. C. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc., 130 (2002), 3585-3595. doi: 10.1090/S0002-9939-02-06697-2.
[5]	C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719.
[6]	C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020.
[7]	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.
[8]	A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
[9]	S. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 181-194. doi: 10.1007/s00030-009-0047-4.
[10]	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.
[11]	C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.
[12]	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.
[13]	X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, II, Kinet. Relat. Models, 7 (2014), 291-304. doi: 10.3934/krm.2014.7.291.
[14]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.
[15]	I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012.
[16]	G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.
[17]	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.
[18]	P. Penel and M. Pokorný, On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353. doi: 10.1007/s00021-010-0038-6.
[19]	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.
[20]	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.
[21]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.
[22]	J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.
[23]	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.
[24]	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.
[25]	K. Yamazaki, Regularity criteria of porous media equation in terms of one partial derivative or pressure field, Commun. Math. Sci., to appear.
[26]	K. Yamazaki, Regularity criteria of supercritical beta-generalized quasi-geostrophic equation in terms of partial derivatives, Electron. J. Differential Equations, 2013 (2013), 1-12.
[27]	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.
[28]	K. Yamazaki, Regularity criteria of MHD system involving one velocity component and one current density component, J. Math. Fluid Mech., 16 (2014), 551-570. doi: 10.1007/s00021-014-0178-1.
[29]	K. Yamazaki, Remarks on the regularity criteria of three-dimensional magnetohydrodynamics system in terms of two velocity field components, J. Math. Phys., 55 (2014), 031505, 16pp. doi: 10.1063/1.4868277.
[30]	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.
[31]	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.
[32]	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.
[33]	Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886. doi: 10.3934/dcds.2005.12.881.
[34]	Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbbR^{3}$, Proc. Amer. Math. Soc., 134 (2006), 149-156. doi: 10.1090/S0002-9939-05-08312-7.
[35]	Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014.

[1]	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
[2]	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
[3]	Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6339-6357. doi: 10.3934/dcdsb.2021021
[4]	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
[5]	Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048
[6]	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 and Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583
[7]	Xin Zhong. Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum. Communications on Pure and Applied Analysis, 2022, 21 (2) : 493-515. doi: 10.3934/cpaa.2021185
[8]	Xin Zhong. Global well-posedness to the nonhomogeneous magneto-micropolar fluid equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022102
[9]	Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6017-6026. doi: 10.3934/dcdsb.2020377
[10]	Yang Liu, Nan Zhou, Renying Guo. Global solvability to the 3D incompressible magneto-micropolar system with vacuum. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022061
[11]	Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083
[12]	Joelma Azevedo, Claudio Cuevas, Jarbas Dantas, Clessius Silva. On the fractional chemotaxis Navier-Stokes system in the critical spaces. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022088
[13]	Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[14]	Vena Pearl Bongolan-walsh, David Cheban, Jinqiao Duan. Recurrent motions in the nonautonomous Navier-Stokes system. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 255-262. doi: 10.3934/dcdsb.2003.3.255
[15]	Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 655-673. doi: 10.3934/dcdsb.2010.14.655
[16]	Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033
[17]	Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039
[18]	Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008
[19]	Igor Kukavica. Interior gradient bounds for the 2D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 873-882. doi: 10.3934/dcds.2001.7.873
[20]	Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471

2021 Impact Factor: 1.398

Tools

Metrics

Other articles
by authors

on AIMS
Kazuo Yamazaki
on Google Scholar
Kazuo Yamazaki

[Back to Top]