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September  2020, 19(9): 4455-4478. doi: 10.3934/cpaa.2020203

Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system

Bin Han 1, and  Na Zhao 2,,

1. 

Department of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, China
2. 

Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

* Corresponding author

Received  November 2019 Revised  March 2020 Published  June 2020

Fund Project: The first author is supported by NSFC grant 11701131. The second author is supported by China Postdoctoral Science Foundation grant 2019TQ0042

In this work, we study the regularity criterion for the 3D incompressible MHD equations. By making use of the structure of the system, we obtain a criterion that is imposed on the magnetic vector field and only one component of the velocity vector field, both in scaling invariant spaces. Moreover, the norms imposed on the magnetic vector field are the Lebesgue and anisotropic Lebesgue norms. This improved the result of our previous blow up criterion in [15], in which the magnetic vector field is bounded in critical Sobolev spaces.

Keywords: Blow up criterion, three dimensional, incompressible, magnetohydrodynamics, Littlewood–Paley decomposition.
Mathematics Subject Classification: Primary:35Q35;Secondary:76D03.
Citation: Bin Han, Na Zhao. Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4455-4478. doi: 10.3934/cpaa.2020203
References:
[1]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. R. Soc. Edinb. Sect. A, 138 (2008), 447-476.  doi: 10.1017/S0308210506001181.

[2]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften 343, Springer-Verlag Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Ec. Norm. Super., 14 (1981), 209-246. 

[4]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differ. Equ., 254 (2013), 2661-2681.  doi: 10.1016/j.jde.2013.01.002.

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differ. Equ., 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.

[6]

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.

[7]

J. Y. Chemin, D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.  doi: 10.1016/j.aim.2015.09.004.

[8]

J. Y. Chemin, M. Paicu and P. Zhang, Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable, J. Differ. Equ., 256 (2014), 223-252.  doi: 10.1016/j.jde.2013.09.004.

[9]

J. Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Commun. Math. Phys., 272 (2007), 529-566.  doi: 10.1007/s00220-007-0236-0.

[10]

J. Y. Chemin and P. Zhang, On the critical one component regularity for 3-D Navier-Stokes system, Ann. Ec. Norm. Super., 49 (2016), 131-167.  doi: 10.24033/asens.2278.

[11]

J. Y. Chemin, P. Zhang and Z. Zhang, On the critical one component regularity for 3-D Navier-Stokes system: general case, Arch. Ration. Mech. Anal., 224 (2017), 871-905.  doi: 10.1007/s00205-017-1089-0.

[12]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.

[13]

X. X. Guo, Y. Du and P. Lu, The regularity criteria on the magnetic field to the 3D incompressible MHD equations, Commun. Math. Sci., 17 (2019), 2257-2280. 

[14]

B. Han, Z. Lei, D. Li and N. Zhao, Sharp one component regularity for Navier–Stokes, Arch. Ration. Mech. Anal., 231 (2019), 939-970.  doi: 10.1007/s00205-018-1292-7.

[15]

B. Han and N. Zhao, On the critical blow up criterion with one velocity component for 3D incompressible MHD system, Nonlinear Anal. Real World Appl., 51 (2020), Art. 103000. doi: 10.1016/j.nonrwa.2019.103000.

[16]

D. Li, On Kato–Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23-100.  doi: 10.4171/rmi/1049.

[17]

F. Lin, Some analytical issues for elastic complex fluids, Commun. Pure Appl. Math., 65 (2012), 893-919.  doi: 10.1002/cpa.21402.

[18]

Y. Liu, On the critical one-component velocity regularity criteria to 3-D incompressible MHD system, J. Differ. Equ., 260 (2016), 6989-7019.  doi: 10.1016/j.jde.2016.01.023.

[19]

M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 21 (2005), 179-235.  doi: 10.4171/RMI/420.

[20]

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

[21]

H. Wang, Y. Li, Z. G. Guo and Z. Skalak, Conditional regularity for the 3D incompressible MHD equations via partial components, Commun. Math. Sci., 17 (2019), 1025-1043.  doi: 10.4310/CMS.2019.v17.n4.a8.

[22]

K. Yamazaki, On the three-dimensional magnetohydrodynamics system in scaling-invariant spaces, Bull. Sci. Math., 140 (2016), 575-614.  doi: 10.1016/j.bulsci.2015.08.003.

[23]

K. Yamazaki, Component reduction for regularity criteria of the three-dimensional magnetohydrodynamics systems, Electron. J. Differ. Equ., (2014), 18 pp.

[24]

K. Yamazaki, Regularity criteria of MHD system involving one velocity and one current density component, J. Math. Fluid Mech., 16 (2014), 551-570.  doi: 10.1007/s00021-014-0178-1.

[25]

K. Yamazaki, Remarks on the regularity criteria of the three-dimensional magnetohydrodynamics system in terms of two velocity field components, J. Math. Phys., 55 (2014), Art. 031505, 16 pp. doi: 10.1063/1.4868277.

[26]

K. Yamazaki, Regularity criteria of the three-dimensional MHD system involving one velocity and one vorticity component, Nonlinear Anal., 135 (2016), 835-846.  doi: 10.1016/j.na.2016.01.015.

[27]

Z. Zhang, Remarks on the global regularity criteria for the 3D MHD equations via two components, Z. Angew. Math. Phys., 66 (2015), 977-987.  doi: 10.1007/s00033-014-0461-2.

show all references

References:
[1]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. R. Soc. Edinb. Sect. A, 138 (2008), 447-476.  doi: 10.1017/S0308210506001181.

[2]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften 343, Springer-Verlag Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Ec. Norm. Super., 14 (1981), 209-246. 

[4]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differ. Equ., 254 (2013), 2661-2681.  doi: 10.1016/j.jde.2013.01.002.

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differ. Equ., 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.

[6]

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.

[7]

J. Y. Chemin, D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.  doi: 10.1016/j.aim.2015.09.004.

[8]

J. Y. Chemin, M. Paicu and P. Zhang, Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable, J. Differ. Equ., 256 (2014), 223-252.  doi: 10.1016/j.jde.2013.09.004.

[9]

J. Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Commun. Math. Phys., 272 (2007), 529-566.  doi: 10.1007/s00220-007-0236-0.

[10]

J. Y. Chemin and P. Zhang, On the critical one component regularity for 3-D Navier-Stokes system, Ann. Ec. Norm. Super., 49 (2016), 131-167.  doi: 10.24033/asens.2278.

[11]

J. Y. Chemin, P. Zhang and Z. Zhang, On the critical one component regularity for 3-D Navier-Stokes system: general case, Arch. Ration. Mech. Anal., 224 (2017), 871-905.  doi: 10.1007/s00205-017-1089-0.

[12]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.

[13]

X. X. Guo, Y. Du and P. Lu, The regularity criteria on the magnetic field to the 3D incompressible MHD equations, Commun. Math. Sci., 17 (2019), 2257-2280. 

[14]

B. Han, Z. Lei, D. Li and N. Zhao, Sharp one component regularity for Navier–Stokes, Arch. Ration. Mech. Anal., 231 (2019), 939-970.  doi: 10.1007/s00205-018-1292-7.

[15]

B. Han and N. Zhao, On the critical blow up criterion with one velocity component for 3D incompressible MHD system, Nonlinear Anal. Real World Appl., 51 (2020), Art. 103000. doi: 10.1016/j.nonrwa.2019.103000.

[16]

D. Li, On Kato–Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23-100.  doi: 10.4171/rmi/1049.

[17]

F. Lin, Some analytical issues for elastic complex fluids, Commun. Pure Appl. Math., 65 (2012), 893-919.  doi: 10.1002/cpa.21402.

[18]

Y. Liu, On the critical one-component velocity regularity criteria to 3-D incompressible MHD system, J. Differ. Equ., 260 (2016), 6989-7019.  doi: 10.1016/j.jde.2016.01.023.

[19]

M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 21 (2005), 179-235.  doi: 10.4171/RMI/420.

[20]

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

[21]

H. Wang, Y. Li, Z. G. Guo and Z. Skalak, Conditional regularity for the 3D incompressible MHD equations via partial components, Commun. Math. Sci., 17 (2019), 1025-1043.  doi: 10.4310/CMS.2019.v17.n4.a8.

[22]

K. Yamazaki, On the three-dimensional magnetohydrodynamics system in scaling-invariant spaces, Bull. Sci. Math., 140 (2016), 575-614.  doi: 10.1016/j.bulsci.2015.08.003.

[23]

K. Yamazaki, Component reduction for regularity criteria of the three-dimensional magnetohydrodynamics systems, Electron. J. Differ. Equ., (2014), 18 pp.

[24]

K. Yamazaki, Regularity criteria of MHD system involving one velocity and one current density component, J. Math. Fluid Mech., 16 (2014), 551-570.  doi: 10.1007/s00021-014-0178-1.

[25]

K. Yamazaki, Remarks on the regularity criteria of the three-dimensional magnetohydrodynamics system in terms of two velocity field components, J. Math. Phys., 55 (2014), Art. 031505, 16 pp. doi: 10.1063/1.4868277.

[26]

K. Yamazaki, Regularity criteria of the three-dimensional MHD system involving one velocity and one vorticity component, Nonlinear Anal., 135 (2016), 835-846.  doi: 10.1016/j.na.2016.01.015.

[27]

Z. Zhang, Remarks on the global regularity criteria for the 3D MHD equations via two components, Z. Angew. Math. Phys., 66 (2015), 977-987.  doi: 10.1007/s00033-014-0461-2.

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