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Regularity criteria for the incompressible Hall-magnetohydrodynamic equations. (English) Zbl 1297.35067
Summary: We establish some new regularity criteria for the three-dimensional incompressible Hall-magnetohydrodynamic equations.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35Q35 PDEs in connection with fluid mechanics 76W05 Magnetohydrodynamics and electrohydrodynamics 70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
Besov space
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##### References:
 [1] Lighthill, M. J., Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Philos. Trans. R. Soc. Lond. Ser. A, 252, 397-430, (1960) · Zbl 0097.20806 [2] Polygiannakis, J. M.; Moussas, X., A review of magneto-vorticity induction in Hall-MHD plasmas, Plasma Phys. Control. Fusion, 43, 195-221, (2001) [3] Fan, J.; Jiang, S.; Nakamura, G.; Zhou, Y., Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech., 13, 557-571, (2011) · Zbl 1270.35339 [4] Zhou, Y.; Fan, J., Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24, 691-708, (2012) · Zbl 1247.35115 [5] Chen, Q.; Miao, C.; Zhang, Z., On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamic equations, Comm. Math. Phys., 284, 919-930, (2008) · Zbl 1168.35035 [6] Lei, Z.; Zhou, Y., BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25, 2, 575-583, (2009) · Zbl 1171.35452 [7] Zhou, Y., Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41, 1174-1180, (2006) · Zbl 1160.35506 [8] Acheritogaray, M.; Degond, P.; Frouvelle, A.; Liu, J.-G., Kinetic formulation and global existence for the Hall-magnetohydrodynamics system, Kinet. Relat. Models, 4, 901-918, (2011) · Zbl 1251.35076 [9] Chae, D.; Lee, J., On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations, 256, 11, 3835-3858, (2014) · Zbl 1295.35122 [10] Chae, D.; Schonbek, M., On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255, 11, 3971-3982, (2013) · Zbl 1291.35212 [11] Chae, D.; Degond, P.; Liu, J.-G., Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31, 555-565, (2014) · Zbl 1297.35064 [12] D. Chae, S. Weng, Singularity formation for the incompressible Hall-MHD equations without resistivity, 19 December 2013, arXiv:1312.5519v1 [math. AP]. · Zbl 1347.35199 [13] Dumas, E.; Sueur, F., On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations, Comm. Math. Phys., 330, 1179-1225, (2014) · Zbl 1294.35094 [14] Dreher, J.; Runban, V.; Grauer, R., Axisymmetric flows in Hall-MHD: a tendency towards finite-time singularity formation, Phys. Scr., 72, 451-455, (2005) [15] Fan, J.; Huang, S.; Nakamura, G., Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations, Appl. Math. Lett., 26, 9, 963-967, (2013) · Zbl 1315.35164 [16] Fan, J.; Ozawa, T., Regularity criteria for Hall-magnetohydrodynamics and the space-time monopole equation in Lorenz gauge, Contemp. Math., 612, 81-89, (2014) · Zbl 1297.35068 [17] Triebel, H., Theory of function spaces, (1983), Birkhäuser Basel · Zbl 0546.46028 [18] Chen, Q.; Miao, C., Existence theorem and blow-up criterion of the strong solutions to the two-fluid MHD equation in $$\mathbb{R}^3$$, J. Differential Equations, 239, 251-271, (2007) · Zbl 1267.76130 [19] Ogawa, T., Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34, 1318-1330, (2003) · Zbl 1036.35082
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