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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
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