GLOBAL REGULARITY FOR 3D GENERALIZED HALL MAGNETO-HYDRODYNAMICS EQUATIONS

Baoquan Yuany; Chaoying Li

doi:10.11948/2018.1143

2018 Volume 8 Issue 4

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Baoquan Yuany, Chaoying Li. GLOBAL REGULARITY FOR 3D GENERALIZED HALL MAGNETO-HYDRODYNAMICS EQUATIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1143-1158. doi: 10.11948/2018.1143

Citation:

Baoquan Yuany, Chaoying Li. GLOBAL REGULARITY FOR 3D GENERALIZED HALL MAGNETO-HYDRODYNAMICS EQUATIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1143-1158. doi: 10.11948/2018.1143

GLOBAL REGULARITY FOR 3D GENERALIZED HALL MAGNETO-HYDRODYNAMICS EQUATIONS

School of Mathematics and Information Science, Henan Polytechnic University, Century Avenue, Jiaozuo, Henan, 454000, China

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Abstract

Abstract

For the 3D incompressible Hall magneto-hydrodynamics equations, global regularity of the weak solutions is not established so far. The major difficulty is that the dissipation given by the Laplacian operator is insufficient to control the nonlinearities. Wan obtained the global regularities of the 3D generalized Hall-MHD equations with critical and subcritical hyperdissipation in (Global regularity for generalized Hall-magnetohydrodynamics systems, Electron. J. Differential Equations, 2015, 2015(179), 1-18). We improve this slightly by making logarithmic reductions in the dissipation and still obtain the global regularity.
- Hall magneto-hydrodynamics equations /
- global regularity /
- hyperdissipation /
- Littlewood-Paley decomposition
MSC: 35Q35;35B65

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References

[1]	M. Acheritogaray, P. Degond, A. Frouvelle and J. G. Liu, Kinetic formulation and global existence for the Hall-Magnetohydrodynamics system, Kinet. Relat. Models, 2011, 4(4), 901-918. Google Scholar
[2]	H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer Heidelberg Dordrecht London New York, Springer-Verlag Berlin Heidelberg, 2011. Google Scholar
[3]	J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Google Scholar
[4]	D. Barbato, F. Morandin and M. Romito, Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system, Anal. PDE, 2014, 7(8), 2009-2027. Google Scholar
[5]	S. A. Balbus and C. Terquem, Linear analysis of the Hall effect in protostellar disks, Astrophysical Journal, 2001, 552(1), 235-247. Google Scholar
[6]	D. F. Bian and B. Yuan, Regularity of weak solutions to the generalized NavierStokes equations (Chinese), Acta Math. Sci. Ser. A (Chin. Ed.), 2011, 31(6), 1601-1609. Google Scholar
[7]	J. Y. Chemin, Perfect Incompressible Fluids, Clarendon press, Oxford, 1998. Google Scholar
[8]	D. Chae, P. Degond and J. G. Liu, Well-posedness for Hallmagnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2014, 31(3), 555-565. Google Scholar
[9]	D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-Magnetohydrodynamics, J. Differential Equations, 2014, 256(11), 3835-3858. Google Scholar
[10]	D. Chae, R. H. Wan and J. H. Wu, Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion, J. Math. Fluid Mech., 2015, 17(4), 627-638. Google Scholar
[11]	T. G. Forbes, Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 1991, 62(1-4), 15-36. Google Scholar
[12]	J. S. Fan, Y. Fukumoto, G. Nakamura and Y. Zhou, Regularity criteria for the incompressible Hall-MHD system, ZAMM Z. Angew. Math. Mech., 2015, 95(11), 1156-1160. Google Scholar
[13]	J. S. Fan, F. C. Li and G. Nakamura, Regularity criteria for the incompressible Hall-magnetohydrodynamic equations, Nonlinear Anal., 2014, 109, 173-179. Google Scholar
[14]	M. G. Fei and Z. Y. Xiang, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics with horizontal dissipation, J. Math. Phys., 2015, 56(5), 051504. Google Scholar
[15]	L. Grafakos, Modern Fourier Analysis, Springer-Verlag, New York, 2009. Google Scholar
[16]	F. Y. 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., 2016, 32, 35-51. Google Scholar
[17]	T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for EulerBoussinesq system with critical dissipation, Comm. Partial Differential Equations, 2010, 36(3), 420-445. Google Scholar
[18]	M. J. Lighthill, Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Philos. Trans. R. Soc. Lond. Ser. A, 1960, 252(1014), 397-430. Google Scholar
[19]	N. N. Pan and M. X. Zhu, A new regularity criterion for the 3D generalized Hall-MHD system with β ∈(1/2,1], J. Math. Anal. Appl., 2017, 445(1), 604-611. Google Scholar
[20]	D. A. Shalybkov and V. A. Urpin, The Hall effect and the decay of magnetic fields, Astron. Astrophys., 1997, 321, 685-690. Google Scholar
[21]	H. Triebel, Theory of function spaces, monographs in mathematics, Birkhäuser Verlag, Basel, 1983. Google Scholar
[22]	T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2009, 2(3), 361-366. Google Scholar
[23]	R. H. Wan, Global regularity for generalized Hall-magnetohydrodynamics systems, Electron. J. Differential Equations, 2015, 2015(179), 1-18. Google Scholar
[24]	J. H. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 2011, 13(2), 295-305. Google Scholar
[25]	X. Wu, Y. H. Yu and Y. B. Tang, Global existence and asymptotic behavior for the 3D generalized Hall-MHD system, Nonlinear Anal., 2017, 151, 41-50. Google Scholar
[26]	R. H. Wan and Y. Zhou, On global existence, energy decay and blow-up criteria for the Hall-MHD system, J. Differential Equations, 2015, 259(11), 5982-6008. Google Scholar
[27]	Y. Z. Wang and W. B. Zuo, On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity, Commun. Pure Appl. Anal., 2014, 13(3), 1327-1336. Google Scholar
[28]	R. H. Wan and Y. Zhou, Low regularity well-posedness for the 3D generalized Hall-MHD system, Acta Appl. Math., 2017, 147(1), 95-111. Google Scholar
[29]	Z. Ye, Regularity criteria and small data global existence to the generalized viscous Hall-magnetohydrodynamics, Comput. Math. Appl., 2015, 70(8), 2137-2154. Google Scholar
[30]	B. Q. Yuan and K. H. Liu, The blow-up criterion via horizontal component of velocity for the Hall-MHD equations, Appl. Anal., 2016, 95(11), 2578-2589. Google Scholar
[31]	Z. Ye and Z. J. Zhang, A remark on regularity criterion for the 3D Hall-MHD equations based on the vorticity, Appl. Math. Comput., 2017, 301, 70-77. Google Scholar
[32]	Z. Zhang, A remark on the blow up criterion for the 3D Hall-MHD system in Besov spaces, J. Math. Anal. Appl., 2016, 441(2), 692-701. Google Scholar