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A remark on regularity criterion for the 3D Hall-MHD equations based on the vorticity. (English) Zbl 1411.35232
Summary: In this paper we investigate the regularity criterion for the local-in-time classical solution to the three-dimensional (3D) incompressible Hall-magnetohydrodynamic equations (Hall-MHD). It is proved that the control of the vorticity alone can ensure the smoothness of the solution.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
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-hydrodynamics system, 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] Beale, J.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3-d Euler equations, Commun. Math. Phys., 94, 61-66, (1984) · Zbl 0573.76029
[4] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure Appl. Math., 35, 771-831, (1982) · Zbl 0509.35067
[5] Campos, L., On hydromagnetic waves in atmospheres with application to the Sun, Theor. Comput. Fluid Dyn., 10, 37-70, (1998) · Zbl 0911.76099
[6] Cao, C.; Wu, J., Two regularity criteria for the 3d MHD equations, J. Differ. Equ., 248, 2263-2274, (2010) · Zbl 1190.35046
[7] Cao, C.; Titi, E., Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57, 2643-2661, (2008) · Zbl 1159.35053
[8] Cao, C.; Titi, E., Global regularity criterion for the 3d Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202, 919-932, (2011) · Zbl 1256.35051
[9] Chae, D.; Degond, P.; Liu, J., Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 31, 555-565, (2014) · Zbl 1297.35064
[10] Chae, D.; Lee, J., On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differ. Equ., 256, 3835-3858, (2014) · Zbl 1295.35122
[11] 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
[12] 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
[13] Chen, Q.; Miao, C.; Zhang, Z., The beale-Kato-Majda criterion for the 3d magneto-hydrodynamics equations, Commun. Math. Phys., 275, 861-872, (2007) · Zbl 1138.76066
[14] Clavin, P., Instabilities and nonlinear patterns of overdriven detonations in gases, (Berestycki, H.; Pomeau, Y., Nonlinear PDEs in Condensed Matter and Reactive Flows, (2002), Kluwer), 49-97 · Zbl 1271.76102
[15] Dong, B.; Chen, Z., Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 50, 10, 13, (2009) · Zbl 1283.76016
[16] Droniou, J.; Imbert, C., Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182, 299-331, (2006) · Zbl 1111.35144
[17] Fan, J.; Li, F.; Nakamura, G., Regularity criteria for the incompressible Hall-magnetohydrodynamic equations, Nonlinear Anal., 109, 173-179, (2014) · Zbl 1297.35067
[18] Forbes, T., Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 62, 15-36, (1991)
[19] Giga, Y., Solutions for semilinear parabolic equations in l^p and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equ., 62, 186-212, (1986) · Zbl 0577.35058
[20] He, C.; Xin, Z., On the regularity of solutions to the magnetohydrodynamic equations, J. Differ. Equ., 213, 235-254, (2005) · Zbl 1072.35154
[21] He, F.; Ahmadb, B.; Hayatc, T.; Zhou, Y., On regularity criteria for the 3d Hall-MHD equations in terms of the velocity, Nonlinear Anal. Real World Appl., 32, 35-51, (2016)
[22] Hou, T.; Li, C., Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl, Commun. Pure Appl. Math., 61, 661-697, (2008) · Zbl 1138.35077
[23] Kozono, H.; Ogawa, T.; Taniuchi, Y., The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242, 251-278, (2002) · Zbl 1055.35087
[24] 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
[25] Miao, C.; Yuan, B.; Zhang, B., Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68, 461-484, (2008) · Zbl 1132.35047
[26] Shalybkov, D.; Urpin, V., The Hall effect and the decay of magnetic fields, Astron. Astrophys., 321, 685-690, (1997)
[27] Wan, R.; Zhou, Y., On the global existence, energy decay and blow up criterions for the Hall-MHD system, J. Differ. Equ., 259, 5982-6008, (2015) · Zbl 1328.35185
[28] Wu, J., Regularity criteria for the generalized MHD equations, Commun. Partial Differ. Equ., 33, 285-306, (2008) · Zbl 1134.76068
[29] Ye, Z., Regularity criteria and small data global existence to the generalized viscous Hall-magnetohydrodynamics, Comput. Math. Appl., 70, 2137-2154, (2015)
[30] Zhang, Z., A remark on the blow-up criterion for the 3d Hall-MHD system in Besov spaces, J. Math. Anal. Appl., 441, 692-701, (2016) · Zbl 1338.35077
[31] Zhou, Y., Remarks on regularities for the 3d MHD equations, Discret. Contin. Dyn. Syst., 12, 881-886, (2005) · Zbl 1068.35117
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