# zbMATH — the first resource for mathematics

Global well-posedness for the 3D incompressible Hall-magnetohydrodynamic equations with Fujita-Kato type initial data. (English) Zbl 1414.35177
Summary: Hall-magnetohydrodynamic (Hall-MHD) equations which can be derived from two fluids model or kinetic models [M. Acheritogaray et al., Kinet. Relat. Models 4, No. 4, 901–918 (2011; Zbl 1251.35076)] 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 D. Chae and J. Lee [J. Differ. Equations 256, No. 11, 3835–3858 (2014; Zbl 1295.35122), Remark 2]. 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.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs 76W05 Magnetohydrodynamics and electrohydrodynamics 35B44 Blow-up in context of PDEs
##### Keywords:
Hall-MHD equations; Fujita-Kato; global well-posedness
Full Text:
##### References:
 [1] 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) · Zbl 1251.35076 [2] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (2011) · Zbl 1227.35004 [3] Chae, D.; Degond, P.; Liu, J-G, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31, 555-565, (2014) · Zbl 1297.35064 [4] 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) · Zbl 1295.35122 [5] Chae, D.; Schonbek, M., On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differ. Equ., 255, 3971-3982, (2013) · Zbl 1291.35212 [6] 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) · Zbl 1327.35314 [7] 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) · Zbl 1347.35199 [8] 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) · Zbl 1294.35094 [9] Fan, J.; Huang, S.; Nakamura, G., Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations, Appl. Math. Lett., 26, 963-967, (2013) · Zbl 1315.35164 [10] Fan, J.; Ozawa, T., Regularity criteria for Hall-magnetohydrodynamics and the space-time monopole equation in Lorenz gauge, Contemp. Math., 612, 81-89, (2014) · Zbl 1297.35068 [11] Fan, J.; Li, F.; Nakamura, G., Regularity criteria for the incompressible Hall-magnetohydrodynamic equations, Nonlinear Anal., 109, 173-179, (2014) · Zbl 1297.35067 [12] 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) · Zbl 1304.35539 [13] 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) · Zbl 1321.35087 [14] 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) · Zbl 1410.35113 [15] 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) · Zbl 1342.35248 [16] Forbes, TG, Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 62, 15-36, (1991) [17] 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) · Zbl 1362.35065 [18] Fujita, H.; Kato, T., On the Navier-Stokes initial value problem. I., Arch. Rational Mech. Anal., 16, 269-315, (1964) · Zbl 0126.42301 [19] Homann, H.; Grauer, R., Bifurcation analysis of magneti creconnection in Hall-MHD systems, Phys. D, 208, 59-72, (2005) · Zbl 1154.76392 [20] Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, UK (2001) · Zbl 0983.76001 [21] Mininni, PD; Gómez, DO; Mahajan, SM, Dynamo action in magnetohydrodynamics and Hall magnetohydrody-namics, Astrophys. J., 587, 472-481, (2003) [22] 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) · Zbl 0737.35102 [23] Shalybkov, DA; Urpin, VA, The Hall effect and the decay of magnetic fields, Astron. Astrophys., 321, 685-690, (1997) [24] Wan, R., Global regularity for generalized Hall-MHD system, Electron. J. Differ. Equ., 179, 1-18, (2015) [25] Wan, R.; Zhou, Y., Low regularity well-posedness for the 3D generalized Hall-MHD system, Acta Appl. Math., 147, 95-111, (2017) · Zbl 1365.35219 [26] 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) · Zbl 1328.35185 [27] Wardle, M., Star formation and the Hall effect, Astrophys. Space Sci., 292, 317-323, (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.