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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 & 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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