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

Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system

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.

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. CaoD. 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. CheminD. S. McCormickJ. 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. CheminM. 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. CheminP. 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. GuoY. 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. HanZ. LeiD. 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. WangY. LiZ. 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. CaoD. 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. CheminD. S. McCormickJ. 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. CheminM. 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. CheminP. 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. GuoY. 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. HanZ. LeiD. 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. WangY. LiZ. 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.

[1]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3045-3062. doi: 10.3934/dcds.2020397

[2]

Xin Zhong. A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3249-3264. doi: 10.3934/dcdsb.2018318

[3]

Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations and Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007

[4]

Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327

[5]

Anthony Suen. Corrigendum: A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1387-1390. doi: 10.3934/dcds.2015.35.1387

[6]

Anthony Suen. A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3791-3805. doi: 10.3934/dcds.2013.33.3791

[7]

Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009

[8]

Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115

[9]

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

[10]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639

[11]

Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167

[12]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure and Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225

[13]

Yue Cao. Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities. Electronic Research Archive, 2020, 28 (1) : 27-46. doi: 10.3934/era.2020003

[14]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[15]

Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure and Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923

[16]

Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic and Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043

[17]

Jeongho Kim, Weiyuan Zou. Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain. Kinetic and Related Models, 2020, 13 (3) : 623-651. doi: 10.3934/krm.2020021

[18]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333

[19]

Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239

[20]

Hao Chen, Kaitai Li, Yuchuan Chu, Zhiqiang Chen, Yiren Yang. A dimension splitting and characteristic projection method for three-dimensional incompressible flow. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 127-147. doi: 10.3934/dcdsb.2018111

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (215)
  • HTML views (84)
  • Cited by (0)

Other articles
by authors

[Back to Top]