Applications of Mathematics, first online, pp. 1-10


An improved regularity criteria for the MHD system based on two components of the solution

Zujin Zhang, Yali Zhang

Received October 4, 2019.   Published online December 4, 2020.

Abstract:  As observed by Yamazaki, the third component $b_3$ of the magnetic field can be estimated by the corresponding component $u_3$ of the velocity field in $L^{\lambda}$ $(2\leq\lambda\leq6)$ norm. This leads him to establish regularity criterion involving $u_3, j_3$ or $u_3,\omega_3$. Noticing that $\lambda$ can be greater than 6 in this paper, we can improve previous results.
Keywords:  MHD equations; regularity criteria
Classification MSC:  35B65, 35Q35, 76D03
DOI:  10.21136/AM.2020.0259-19

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References:
[1] C. Cao, J. Wu: Two regularity criteria for the 3D MHD equations. J. Differ. Equations 248 (2010), 2263-2274. DOI 10.1016/j.jde.2009.09.020 | MR 2595721 | Zbl 1190.35046
[2] J.-Y. Chemin, P. Zhang: On the critical one component regularity for 3-D Navier-Stokes systems. Ann. Sci. Éc. Norm. Supér. 49 (2016), 131-167. DOI 10.24033/asens.2278 | MR 3465978 | Zbl 1342.35210
[3] Q. Chen, C. Miao, Z. Zhang: On the regularity criterion of weak solutions for the 3D viscous magneto-hydrodynamics equations. Commun. Math. Phys. 284 (2008), 919-930. DOI 10.1007/s00220-008-0545-y | MR 2452599 | Zbl 1168.35035
[4] H. Duan: On regularity criteria in terms of pressure for the 3D viscous MHD equations. Appl. Anal. 91 (2012), 947-952. DOI 10.1080/00036811.2011.556626 | MR 2911243 | Zbl 1247.35101
[5] G. Duvaut, J. L. Lions: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46 (1972), 241-279. (In French.) DOI 10.1007/BF00250512 | MR 0346289 | Zbl 0264.73027
[6] C. He, Z. Xin: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equations 213 (2005), 235-254. DOI 10.1016/j.jde.2004.07.002 | MR 2142366 | Zbl 1072.35154
[7] E. Ji, J. Lee: Some regularity criteria for the 3D incompressible magnetohydrodynamics. J. Math. Anal. Appl. 369 (2010), 317-322. DOI 10.1016/j.jmaa.2010.03.015 | MR 2643871 | Zbl 1196.35063
[8] X. Jia: A new scaling invariant regularity criterion for the 3D MHD equations in terms of horizontal gradient of horizontal components. Appl. Math. Lett. 50 (2015), 1-4. DOI 10.1016/j.aml.2015.05.017 | MR 3378940 | Zbl 1327.35320
[9] X. Jia, Y. Zhou: Regularity criteria for the 3D MHD equations involving partial components. Nonlinear Anal., Real World Appl. 13 (2012), 410-418. DOI 10.1016/j.nonrwa.2011.07.055 | MR 2846851 | Zbl 1239.35032
[10] X. Jia, Y. Zhou: Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of $3\times 3$ mixture matrices. Nonlinearity 28 (2015), 3289-3307. DOI 10.1088/0951-7715/28/9/3289 | MR 3403399 | Zbl 1326.35276
[11] X. Jia, Y. Zhou: On regularity criteria for the 3D incompressible MHD equations involving one velocity component. J. Math. Fluid Mech. 18 (2016), 187-206. DOI 10.1007/s00021-015-0246-1 | MR 3461931 | Zbl 1334.35240
[12] L. Ni, Z. Guo, Y. Zhou: Some new regularity criteria for the 3D MHD equations. J. Math. Anal. Appl. 396 (2012), 108-118. DOI 10.1016/j.jmaa.2012.05.076 | MR 2956948 | Zbl 1247.35095
[13] P. Penel, 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 | MR 2824487 | Zbl 1270.35354
[14] M. Sermange, R. Temam: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36 (1983), 635-664. DOI 10.1002/cpa.3160360506 | MR 0716200 | Zbl 0524.76099
[15] 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 | MR 3247368 | Zbl 1307.35237
[16] K. Yamazaki: Remarks on the regularity criteria of three-dimensional magnetohydrodynamics system in terms of two velocity field components. J. Math. Phys. 55 (2014), Article ID 031505, 16 pages. DOI 10.1063/1.4868277 | MR 3221239 | Zbl 1286.76172
[17] 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 | MR 3509263 | Zbl 1345.35081
[18] K. Yamazaki: Regularity criteria of the three-dimensional MHD system involving one velocity and one vorticity component. Nonlinear Anal., Theory Methods Appl., Ser. A 135 (2016), 73-83. DOI 10.1016/j.na.2016.01.015 | MR 3473110 | Zbl 1345.35080
[19] K. Yamazaki: Horizontal Biot-Savart law in general dimension and an application to the 4D magneto-hydrodynamics. Differ. Integral Equ. 31 (2018), 301-328. MR 3738200 | Zbl 06837099
[20] Z. Zhang: Regularity criteria for the 3D MHD equations involving one current density and the gradient of one velocity component. Nonlinear Anal., Theory Methods Appl., Ser. A 115 (2015), 41-49. DOI 10.1016/j.na.2014.12.003 | MR 3305136 | Zbl 1309.35095
[21] 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 | MR 3347420 | Zbl 1329.35097
[22] Z. Zhang: Refined regularity criteria for the MHD system involving only two components of the solution. Appl. Anal. 96 (2017), 2130-2139. DOI 10.1080/00036811.2016.1207245 | MR 3667850 | Zbl 1372.35064
[23] Z. Zhang, Z.-a. Yao, M. Lu, L. Ni: Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations. J. Math. Phys. 52 (2011), Article ID 053103, 7 pages. DOI 10.1063/1.3589966 | MR 2839081 | Zbl 1317.35180
[24] 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 | MR 2128731 | Zbl 1068.35117
[25] Y. Zhou, J. Fan: Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math. 24 (2012), 691-708. DOI 10.1515/form.2011.079 | MR 2949119 | Zbl 1247.35115

Affiliations:   Zujin Zhang (corresponding author), Yali Zhang, School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, Jiangxi, P. R. China, e-mail: zhangzujin361@163.com, 1070469142@qq.com


 
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