A Generalized Blow up Criteria with One Component of Velocity for 3D Incompressible MHD System

Abstract

In this paper, the authors study the global regularity of the 3D magnetohydrodynamics system in terms of one velocity component. In particular, they establish a new Prodi-Serrin type regularity criterion in the framework of weak Lebesgue spaces both in time and space variables.

Keywords

MHD / Regularity / Blow-up criterion

References

[1] Abidi H, Paicu M. Global existence for the magnetohydrodynamic system in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A, 2008, 138: 447-476

[2] Benjamin P, Yu X W. On Prodi-Serrin type conditions for the 3D Navier-Stokes equations. Non-linear Analysis: Theory Methods Applications, 2020, 190: 111612

[3] Cao C, Regmi D, Wu J. The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J. Differential Equations, 2013, 254: 2661-2681

[4] Cao C, Wu J. Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math., 2011, 226: 1803-1822

[5] Chemin J Y, McCormick D S, Robinson J C, Rodrigo J L. Local existence for the non-resistive MHD equations in Besov spaces. Adv. Math., 2016, 286: 1-31

[6] Chemin J Y, Zhang P. On the critical one component regularity for 3-D Navier-Stokes system. Ann. Sci. Ecole. Norm. S., 2016, 49: 131-167

[7] Chemin J Y, Zhang P, Zhang Z. On the critical one component regularity for 3-D Navier-Stokes system: general case. Arch. Ration. Mech. Anal., 2017, 224: 871-905

[8] Duvaut G, Lions J L. Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal., 1972, 46: 241-279

[9] Guo X X, Du Y, Lu P. The regularity criteria on the magnetic field to the 3D incompressible MHD equations. Comm. Math. Sci., 2019, 17: 2257-2280

[10] Han B, Zhao N. Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics. Communications on Pure Applied Analysis, 2020, 19: 4455-4478

[11] Han B, Zhao N. On the critical blow up criterion with one velocity component for 3D incompressible MHD system. Nonlinear Analysis: Real World Applications, 2020, 51: 103000

[12] Ni L D, Guo Z G, Zhou Y. Some new regularity criteria for the 3D MHD equations. J. Math. Anal. Appl., 2012, 396: 108-118

[13] Lemarié-Rieusset P G. Recent Developments in the Navier-Stokes Problem, 2002, Boca Raton: Chapman & Hall/CRC

[14] Wang H, Li Y, Guo Z G, Skalak Z. Conditional regularity for the 3D incompressible MHD equations via partial components. Comm. Math. Sci., 2019, 17: 1025-1043

[15] Yamazaki K. Regularity criteria of MHD system involving one velocity and one current density component. J. Math. Fluid Mech., 2014, 16: 551-570

[16] Yamazaki K. Remarks on the regularity criteria of the three-dimensional magnetohydrodynamics system in terms of two velocity field components. J. Math. Phys., 2014, 55: 031505 16 pp

[17] Yamazaki, K., Component reduction for regularity criteria of the three-dimensional magnetohydrodynamics systems, Electron. J. Differ. Equ., 98, 2014, 18 pp.

[18] Yamazaki K. Regularity criteria of the three-dimensional MHD system involving one velocity and one vorticity component. Nonlinear Anal, 2016, 135: 73-83

[19] Yamazaki K. On the three-dimensional magnetohydrodynamics system in scaling-invariant spaces. Bull. Sci. Math., 2016, 140: 575-614

[20] Zhang Z. Remarks on the global regularity criteria for the 3D MHD equations via two components. Z. Angew. Math. Phys., 2015, 66: 977-987