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AIMS Mathematics

2018, Volume 3, Issue 4: 565-574. doi: 10.3934/Math.2018.4.565
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Research article

A regularity criterion of smooth solution for the 3D viscous Hall-MHD equations

  • 1.
    ENS of Mostaganem, University of Mostaganem, Box 227, Mostaganem 27000, Algeria
  • 2.
    Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6 95125 Catania - Italy
  • 3.
    RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia
  • 4.
    Department of Mathematical Science, Faculty of Applied Science, Umm Alqura University, P. O. Box 14035, Makkah 21955, Saudi Arabia
  • Received: 20 August 2018 Accepted: 20 October 2018 Published: 28 November 2018

  • In this work, we investigate the regularity criterion for the solution of the Hall-MHD system in three-dimensions. It is proved that if the pressure $\pi $ and the gradient of magnetic field $ \nabla B$ satisfies some kind of space-time integrable condition on $[0, T]$, then the corresponding solution keeps smoothness up to time T. This result improves some previous works to the Morrey space $\overset{\cdot }{\mathcal{M }}_{2, \frac{3}{r}}$ for $0\leq r \lt 1$ which is larger than $L^{\frac{3}{r}}$.

    Citation: A. M. Alghamdi, S. Gala, M. A. Ragusa. A regularity criterion of smooth solution for the 3D viscous Hall-MHD equations[J]. AIMS Mathematics, 2018, 3(4): 565-574. doi: 10.3934/Math.2018.4.565

    Related Papers:

  • In this work, we investigate the regularity criterion for the solution of the Hall-MHD system in three-dimensions. It is proved that if the pressure $\pi $ and the gradient of magnetic field $ \nabla B$ satisfies some kind of space-time integrable condition on $[0, T]$, then the corresponding solution keeps smoothness up to time T. This result improves some previous works to the Morrey space $\overset{\cdot }{\mathcal{M }}_{2, \frac{3}{r}}$ for $0\leq r \lt 1$ which is larger than $L^{\frac{3}{r}}$.


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    [1] M. Acheritogaray, P. Degond, A. Frouvelle, et al. Kinetic fomulation and global existence for the Hall-magneto-hydrodynamics system, Kinet. Relat. Mod., 4 (2011), 901–918.
    [2] S. A. Balbus and C. Terquem, Linear analysis of the Hall e_ect in protostellar disks, The Astrophysical Journal, 552 (2001), 235–247.
    [3] J. Bergh and J. L¨ofstrom, Inerpolation Spaces. An Introduction, Springer-Verlag, New York, 1976.
    [4] D. Chae, P. Degond and J. G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. I. H. Poincaré-An, 31 (2014), 555–565
    [5] D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hallmagnetohydrodynamics, J. Di_er. Equations, 256 (2014), 3835–3858.
    [6] D. Chae and M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Di_er. Equations, 255 (2013), 3971–3982.
    [7] D. Chae and S. Weng, Singularity formation for the incompressible Hall-MHD equations without resistivity, Ann. I. H. Poincaré-An, 33 (2016), 1009–1022.
    [8] D. Chae and J. Wolf, On partial regularity for the steady Hall-magnetohydrodynamics system, Commun. Math. Phys., 339 (2015), 1147–1166.
    [9] D. Chae and J. Wolf, On partial regularity for the 3D nonstationary Hall magnetohydrodynamics equations on the plane, SIAM J. Math. Anal., 48 (2016), 443–469.
    [10] J. Fan, Y. Fukumoto, G. Nakamura, et al. Regularity criteria for the incompressible Hall-MHD system, Zamm-Z Angew. Math. Me., 95 (2015), 1156–1160.
    [11] J. Fan, X. Jia, G. Nakamura, et al. On well-posedness and blowup criteria for the
    [12] magnetohydrodynamics with the Hall and ion-slip e_ects, Z. Angew. Math. Phys., 66 (2015), 1695–1706.
    [13] J. Fan, A. Alsaedi, T. Hayat, et al. On strong solutions to the compressible Hallmagnetohydrodynamic system, Nonlinear Anal-Real, 22 (2015), 423–434.
    [14] J. Fan, H. Malaikah, S. Monaquel, et al. Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., 175 (2014), 127–131.
    [15] J. Fan and T. Ozawa, Regularity criteria for the incompressible MHD with the Hall or Ion-Slip e_ects, International Journal of Mathematical Analysis, 9 (2015), 1173–1186.
    [16] J. Fan, F. Li and G. Nakamura, Regularity criteria for the incompressible Hallmagnetohydrodynamic equations, Nonlinear Anal-Theor, 109 (2014), 173–179.
    [17] J. Fan, B. Samet and Y. Zhou, A regularity criterion for a generalized Hall-MHD system, Comput. Math. Appl., 74 (2017), 2438–2443.
    [18] J. Fan, B. Ahmad, T. Hayat, et al. On well-posedness and blow-up for the full compressible Hall-MHD system, Nonlinear Anal-Real, 31 (2016), 569–579.
    [19] M. Fei and Z. Xiang, On the blow-up criterion and small data global existence for the Hallmagnetohydrodynamics with horizontal dissipation, J. Math. Phys., 56 (2015), 051504.
    [20] T. G. Forbes, Magnetic reconnection in solar flares, Geophys. Astro. Fluid, 62 (1991), 15–36.
    [21] S. Gala, Regularity criterion for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space, NoDEA-Nonlinear Di_., 17 (2010), 181–194.
    [22] S. Gala, On the regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space, Nonlinear Anal-Real, 12 (2011), 2142–2150.
    [23] S. Gala, A new regularity criterion for the 3D MHD equations in R3 , Commun. Pur. Appl. Anal., 11 (2012), 973–980.
    [24] J. Geng, X. Chen and S. Gala, On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space, Commun. Pur. Appl. Anal., 10 (2011), 583–592.
    [25] Z. Guo and S. Gala, Remarks on logarithmical regularity criteria for the Navier-Stokes equations, J. Math. Phys., 52 (2011), 063503.
    [26] F. He, B. Ahmad, T. Hayat, et al. On regularity criteria for the 3D Hall-MHD equations in terms of the velocity, Nonlinear Anal-Real, 32 (2016), 35–51.
    [27] X. Jia and Y. Zhou, Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of 3 × 3 mixture matrices, Nonlinearity, 28 (2015), 3289–3307.
    [28] Z. Jiang, Y. Wang and Y. Zhou, On regularity criteria for the 2D generalized MHD system, J. Math. Fluid Mech., 18 (2016), 331–341.
    [29] P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897–930.
    [30] M. J. Lighthill, Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Philos. T. R. Soc. A, 252 (1960), 397–430.
    [31] S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces, Proc. Amer. Math. Soc., 131 (2003), 1553–1556.
    [32] Y. Meyer, P. Gerard and F. Oru, Inégalités de Sobolev précisées, Séminaire sur les équations aux Dérivées Partielles (Polytechnique), 1996 (1997), 1–8.
    [33] D. A. Shalybkov and V. A. Urpin, The Hall e_ect and the decay of magnetic fields, Astronomy and Astrophysics, 321 (1997), 685–690.
    [34] R.Wan and Y. Zhou, On global existence, energy decay and blow up criterions for the Hall-MHD system, J. Di_er. Equations, 259 (2015), 5982–6008.
    [35] Y. Wang and W. Zuo, On the blow-up criterion of smooth solutions for Hallmagnetohydrodynamics system with partial viscosity, Commun. Pur. Appl. Anal., 13 (2014), 1327–1336.
    [36] Z. Ye, Regularity criterion for the 3D Hall-magnetohydrodynamic equations involving the vorticity, Nonlinear Anal-Theor, 144 (2016), 182–193.
    [37] Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field, Nonlinear Anal-Theor, 72 (2010), 3643–3648.
    [38] Y. Zhou and S. Gala, Regularity criteria in terms of the pressure for the Navier-Stokes Equations in the critical Morrey-Campanato space, Z. Anal. Anwend., 30 (2011), 83–93.
    [39] R. Wan and Y. Zhou, Low regularity well-posedness for the 3D generalized Hall-MHD system, Acta Appl. Math., 147 (2017), 95–111.
    [40] R. Wan and Y. Zhou, On global existence, energy decay and blow-up criteria for the Hall-MHD system, J. Di_er. Equations, 259 (2015), 5982–6008.
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