Abstract
Hall-magnetohydrodynamic (Hall-MHD) equations which can be derived from two fluids model or kinetic models [see Acheritogaray et al. (Kinet Relat Models 4:901–918, 2011)] plays a crucial role in the study of magnetic reconnection in space plasmas, star formation, neutron stars. In this paper, we obtain two Fujita–Kato type results for the 3D Hall-MHD equations, which almost give positive answers to the question proposed by Chae and Lee (Remark 2 in Chae and Lee [J Differ Equ 256:3835–3858, 2014)]. The coupling between u and B is the main difficulty. Our idea is splitting the Navier–Stokes equations from the Hall-MHD equations and combining with some suitable blow-up criteria.
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References
Acheritogaray, M., Degond, P., Frouvelle, A., Liu, J.-G.: Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinet. Relat. Models 4, 901–918 (2011)
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (2011)
Chae, D., Degond, P., Liu, J.-G.: Well-posedness for Hall-magnetohydrodynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 555–565 (2014)
Chae, D., Lee, J.: On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics. J. Differ. Equ. 256, 3835–3858 (2014)
Chae, D., Schonbek, M.: On the temporal decay for the Hall-magnetohydrodynamic equations. J. Differ. Equ. 255, 3971–3982 (2013)
Chae, D., Wan, R., Wu, J.: Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion. J. Math. Fluid Mech. 17, 627–638 (2015)
Chae, D., Weng, S.: Singularity formation for the incompressible Hall-MHD equations without resistivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 1009–1022 (2016)
Dumas, E., Sueur, F.: On the weak solutions to the Maxwell–Landau–Lifshitz equations and to the Hall-magnetohydrodynamic equations. Commun. Math. Phys. 330, 1179–1225 (2014)
Fan, J., Huang, S., Nakamura, G.: Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations. Appl. Math. Lett. 26, 963–967 (2013)
Fan, J., Ozawa, T.: Regularity criteria for Hall-magnetohydrodynamics and the space-time monopole equation in Lorenz gauge. Contemp. Math. 612, 81–89 (2014)
Fan, J., Li, F., Nakamura, G.: Regularity criteria for the incompressible Hall-magnetohydrodynamic equations. Nonlinear Anal. 109, 173–179 (2014)
Fan, J., Alsaedi, A., Hayat, T., Nakamura, G., Zhou, Y.: On strong solutions to the compressible Hallmagnetohydrodynamic system. Nonlinear Anal. Real World Appl. 22, 423–434 (2015)
Fan, J., Jia, X., Nakamura, G., Zhou, Y.: On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects. Z. Angew. Math. Phys. 66, 1695–1706 (2015)
Fan, J., Ahmad, B., Hayat, T., Zhou, Y.: On blow-up criteria for a new Hall-MHD system. Appl. Math. Comput. 274, 20–24 (2016)
Fan, J., Ahmad, B., Hayat, T., Zhou, Y.: On well-posedness and blow-up for the full compressible Hall-MHD system. Nonlinear Anal. Real World Appl. 31, 569–579 (2016)
Forbes, T.G.: Magnetic reconnection in solar flares. Geophys. Astrophys. Fluid Dyn. 62, 15–36 (1991)
He, F., Ahmad, B., Hayat, T., Zhou, Y.: On regularity criteria for the 3D Hall-MHD equations in terms of the velocity. Nonlinear Anal. Real World Appl. 32, 35–51 (2016)
Fujita, H., Kato, T.: On the Navier–Stokes initial value problem. I. Arch. Rational Mech. Anal. 16, 269–315 (1964)
Homann, H., Grauer, R.: Bifurcation analysis of magneti creconnection in Hall-MHD systems. Phys. D 208, 59–72 (2005)
Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, UK (2001)
Mininni, P.D., Gómez, D.O., Mahajan, S.M.: Dynamo action in magnetohydrodynamics and Hall magnetohydrody-namics. Astrophys. J. 587, 472–481 (2003)
Kenig, C., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)
Shalybkov, D.A., Urpin, V.A.: The Hall effect and the decay of magnetic fields. Astron. Astrophys. 321, 685–690 (1997)
Wan, R.: Global regularity for generalized Hall-MHD system. Electron. J. Differ. Equ. 179, 1–18 (2015)
Wan, R., Zhou, Y.: Low regularity well-posedness for the 3D generalized Hall-MHD system. Acta Appl. Math. 147, 95–111 (2017)
Wan, R., Zhou, Y.: On global existence, energy decay and blow-up criteria for the Hall-MHD system. J. Differ. Equ. 259, 5982–6008 (2015)
Wardle, M.: Star formation and the Hall effect. Astrophys. Space Sci. 292, 317–323 (2004)
Acknowledgements
The authors thank the referee for the careful reading and helpful comments. Wan was supported by the NSF of the Jiangsu Higher Education Institutions of China (18KJB110018), the NSF of Jiangsu Province (BK20180721). Zhou was supported by NSFC (No. 11171154).
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Wan, R., Zhou, Y. Global Well-Posedness for the 3D Incompressible Hall-Magnetohydrodynamic Equations with Fujita–Kato Type Initial Data. J. Math. Fluid Mech. 21, 5 (2019). https://doi.org/10.1007/s00021-019-0410-0
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DOI: https://doi.org/10.1007/s00021-019-0410-0