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AIMS Mathematics

2016, Volume 1, Issue 3: 282-287. doi: 10.3934/Math.2016.3.282
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Research article

A note on the Liouville type theorem for the smooth solutions of the stationary Hall-MHD system

  • Department of Mathematics, University of Mostaganem, Box 227, Mostaganem 27000, Algeria
  • Received: 25 May 2016 Accepted: 26 August 2016 Published: 13 October 2016

  • The main result of this work is to study the Liouville type theorem for the stationary Hall-MHD system on $\mathbb{R}.{3}$. Specificaly,we show that if $(u,B)$ is a smooth solutions to Hall-MHD equations satisfying $(u,B)\in L.%\frac{9}{2}(\mathbb{R}.3)$,then we have $u=B=0$. This improves a recent result of Chae et al. [2] and Zujin et al. [14].

    Citation: Sadek Gala. A note on the Liouville type theorem for the smooth solutions of the stationary Hall-MHD system[J]. AIMS Mathematics, 2016, 1(3): 282-287. doi: 10.3934/Math.2016.3.282

    Related Papers:

  • The main result of this work is to study the Liouville type theorem for the stationary Hall-MHD system on $\mathbb{R}.{3}$. Specificaly,we show that if $(u,B)$ is a smooth solutions to Hall-MHD equations satisfying $(u,B)\in L.%\frac{9}{2}(\mathbb{R}.3)$,then we have $u=B=0$. This improves a recent result of Chae et al. [2] and Zujin et al. [14].


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    [1] M. Acheritogaray, P. Degond, A. Frouvelle and J.G. Liu, Kinetic fomulation and global existence for the Hall-magnetohydrodynamics system, Kinet. Relat. Models, 4 (2011), 901-918.
    [2] D. Chae, P. Degond and J.G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 31 (2014), 555-565
    [3] D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hallmagnetohydrodynamics, J. Di er. Equ., 256 (2014), 3835-3858.
    [4] J. Fan, A. Alsaedi, T. Hayat, G. Nakamura and Y. Zhou, On strong solutions to the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 22 (2015), 423-434.
    [5] J. Fan, X. Jia, G. Nakamura and Y. Zhou, On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects, Z. Angew. Math. Phys., 66 (2015), no. 4, 1695-1706.
    [6] J. Fan, B. Ahmad, T. Hayat and Y. Zhou, On blow-up criteria for a new Hall-MHD system, Appl. Math. Comput., 274 (2016), 20-24.
    [7] J. Fan, B. Ahmad, T. Hayat and Y. Zhou, On well-posedness and blow-up for the full compressible Hall-MHD system, Nonlinear Anal. Real World Appl., 31 (2016), 569-579.
    [8] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady State Problems. 2nd Edition, Springer Monographs in Mathematics, Springer, NewYork, 2011.
    [9] F. He, B. Ahmad, T. Hayat and Y. Zhou, On regularity criteria for the 3D Hall-MHD equations in terms of the velocity, Nonlinear Anal. Real World Appl., 32 (2016), 35-51.
    [10] Y. Zhuan, Regulatity criterion for the 3D Hall-magnetohydrodynamic equations involing the vorticity, Nonlinear Anal. 144 (2016), 182-193.
    [11] Y. Zhuan, Regulatity criteria and small data global existence to the generalized viscous Hallmagnetohydrodynamics, Comput. Math. Appl., 70 (2015), 2137-2154.
    [12] R. Wan and Y. Zhou, On global existence, energy decay and blow-up criteria for the Hall-MHD system, J. Di erential Equations, 259 (2015), no. 11, 5982-6008.
    [13] R. Wan and Y. Zhou, Yong Low regularity well-posedness for the 3D generalized Hall-MHD system, To appear in Acta Appl. Math., DOI: 10.1007/s10440-016-0070-5.
    [14] Z. Zujin, X. Xian and Q. Shulin, Remarks on Liouville Type Result for the 3D Hall-MHD System, J. Part. Di . Eq., 28 (2015), 286-290.
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  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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