Abstract
In this paper, we investigate the well-posedness results of the three-dimensional incompressible Hall-magnetohydrodynamic equations with fractional dissipation. More precisely, we provide a direct proof of the local well-posedness of smooth solutions for the Hall-magnetohydrodynamic equations with the diffusive term for the magnetic field consisting of the fractional Laplacian with its power bigger than or equal to one half. Furthermore, the small data global well-posedness results are also derived. In addition, we obtain the optimal decay rate when the fractional powers are further restricted to a certain range.
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