3D Hall-MHD system with vorticity in Besov spaces. (English) Zbl 1421.35297

Zhang, Zujin

Ann. Pol. Math. 121, No. 1, 91-98 (2018).

Summary: By introducing some new ideas, using the methods from Z. Ye [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 144, 182–193 (2016; Zbl 1348.35206)] and Z. Ye and Z. Zhang [Appl. Math. Comput. 301, 70–77 (2017; Zbl 1411.35232)] and the result of Z. Zhang [J. Math. Anal. Appl. 441, No. 2, 692–701 (2016; Zbl 1338.35077)], we establish the regularity criterion \[ \boldsymbol{\omega} \in L^{2/s}(0,T;\dot B^s_{\infty ,\infty}(\mathbb R^3)),\;\hskip 1em 0 < s < 1, \] for the \(3\)D Hall-MHD system. This improves several previous results.

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
85A30	Hydrodynamic and hydromagnetic problems in astronomy and astrophysics

Keywords:

Hall-MHD system; regularity criteria; vorticity

References:

[1]

D. A. Shalybkov and V. A. Urpin, The Hall effect and the decay of magnetic fields, Astronomy Astrophys. 321 (1997), 685-690. [15] R. H. Wan and Y. Zhou, Low regularity well-posedness for the 3D generalized HallMHD system, Acta Appl. Math. 147 (2017), 95-111. [16] M. Wardle, Star formation and the Hall effect, Astrophys. Space Sci. 292 (2004), 317-323. [17] Z. Ye, Regularity criterion for the 3D Hall-magnetohydrodynamic equations involving the vorticity, Nonlinear Anal. 144 (2016), 182-193. [18] Z. Ye and Z. J. Zhang, A remark on regularity criterion for the 3D Hall-MHD equations based on the vorticity, Appl. Math. Comput. 301 (2017), 70-77. [19] Z. J. Zhang, A remark on the blow-up criterion for the 3D Hall-MHD system in Besov spaces, J. Math. Anal. Appl. 441 (2016), 692-701. [20] Z. J. Zhang and X. Yang, Navier-Stokes equations with vorticity in Besov spaces of negative regular indices, J. Math. Anal. Appl. 440 (2016), 415-419. Zujin Zhang School of Mathematics and Computer Science Gannan Normal University Ganzhou 341000, Jiangxi, P.R. China E-mail: [e-mail]