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Regularity criterion for the 3D Hall-magnetohydrodynamic equations involving the vorticity. (English) Zbl 1348.35206
The author study the regularity of solutions to the three-dimensional incompressible Hall-magneto-hydrodynamic equations. And the regularity criterion is obtained only in terms of the vorticity fields, which is similar with that of the magneto-hydrodynamic equations.
Reviewer: Cheng He (Beijing)

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
35Q85 PDEs in connection with astronomy and astrophysics
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Acheritogaray, M.; Degond, P.; Frouvelle, A.; Liu, J., Kinetic formulation and global existence for the Hall-magneto-hydrodynamicssystem, Kinet. Relat. Models, 4, 901-918, (2011) · Zbl 1251.35076
[2] Balbus, S.; Terquem, C., Linear analysis of the Hall effect in protostellar disks, Astrophys. J., 552, 235-247, (2001)
[3] Campos, L., On hydromagnetic waves in atmospheres with application to the Sun, Theor. Comput. Fluid Dyn., 10, 37-70, (1998) · Zbl 0911.76099
[4] Chae, D.; Degond, P.; Liu, J., Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31, 555-565, (2014) · Zbl 1297.35064
[5] Chae, D.; Lee, J., On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations, 256, 3835-3858, (2014) · Zbl 1295.35122
[6] Chae, D.; Schonbek, M., On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255, 3971-3982, (2013) · Zbl 1291.35212
[7] Chae, D.; Wan, R.; Wu, J., Local well-posedness for the Hall MHD equations with fractional magnetic diffusion, J. Math. Fluid Mech., 17, 627-638, (2015) · Zbl 1327.35314
[8] Chae, D.; Weng, S., Singularity formation for the incompressible Hall-MHD equations without resistivity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33, 1009-1022, (2016) · Zbl 1347.35199
[9] Chae, D.; Wolf, J., On partial regularity for the steady Hall magnetohydrodynamics system, Comm. Math. Phys., 339, 1147-1166, (2015) · Zbl 1328.35165
[10] Chae, D.; Wolf, J., On partial regularity for the 3D nonstationary Hall magnetohydrodynamics equations on the plane, SIAM J. Math. Anal., 48, 443-469, (2016) · Zbl 1336.35287
[11] Fan, J.; Fukumoto, Y.; Nakamura, G.; Zhou, Y., Regularity criteria for the incompressible Hall-MHD system, ZAMM Z. Angew. Math. Mech., 95, 1156-1160, (2015) · Zbl 1333.35191
[12] Fan, J.; Li, F.; Nakamura, G., Regularity criteria for the incompressible Hall-magnetohydrodynamic equations, Nonlinear Anal., 109, 173-179, (2014) · Zbl 1297.35067
[13] Fei, M.; Xiang, Z., On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics with horizontal dissipation, J. Math. Phys., 56, (2015), 13 pp · Zbl 1317.76093
[14] Forbes, T., Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 62, 15-36, (1991)
[15] He, F.; Ahmadb, B.; Hayatc, T.; Zhou, Y., On regularity criteria for the 3D Hall-MHD equations in terms of the velocity, Nonlinear Anal. RWA, 32, 35-51, (2016)
[16] He, C.; Xin, Z., On the regularity of solutions to the magneto-hydrodynamic equations, J. Differential Equations, 213, 235-254, (2005) · Zbl 1072.35154
[17] Lemarié-Rieusset, P. G., (Recent Developments in the Navier-Stokes Problem, Chapman Hall/CRC Research Notes in Mathematics, vol. 431, (2002), Chapman Hall/CRC Boca Raton, FL) · Zbl 1034.35093
[18] Lighthill, M., Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Philos. Trans. R. Soc. Lond. Ser. A, 252, 397-430, (1960) · Zbl 0097.20806
[19] Shalybkov, D.; Urpin, V., The Hall effect and the decay of magnetic fields, Astronom. Astrophys., 685-690, (1997)
[20] Wan, R.; Zhou, Y., On the global existence, energy decay and blow up criterions for the Hall-MHD system, J. Differential Equations, 259, 5982-6008, (2015) · Zbl 1328.35185
[21] Wang, Y.; Zuo, W., On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity, Commun. Pure Appl. Anal., 13, 1327-1336, (2014) · Zbl 1292.76012
[22] Weng, S., On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differential Equations, 260, 6504-6524, (2016) · Zbl 1341.35133
[23] Weng, S., Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270, 2168-2187, (2016) · Zbl 1347.35207
[24] Ye, Z., Regularity criteria and small data global existence to the generalized viscous Hall-magnetohydrodynamics, Comput. Math. Appl., 70, 2137-2154, (2015)
[25] Zhou, Y., Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12, 881-886, (2005) · Zbl 1068.35117
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