Abstract
In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system for initial data \((u_0,B_0)\in H^{\frac{1}{2}+\sigma }({\mathbb {R}}^3)\times H^{\frac{3}{2}}({\mathbb {R}}^3)\) with \(\sigma \in (0,2)\). In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce \(\sigma \) to 0 with the aid of the new formulation of the Hall-MHD system observed by Danchin and Tan (Commun Partial Differ Equ 46(1):31–65, 2021). Compared with the previous works, our local well-posedness results improve the regularity condition on the initial data. Moreover, we establish the global well-posedness for small initial data in \(H^{\frac{1}{2}+\sigma }({\mathbb {R}}^3)\times H^{\frac{3}{2}}({\mathbb {R}}^3)\) with \(\sigma \in (0,2)\), and get the optimal time-decay rates of solutions.
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Zhang, S. Well-Posedness for the Incompressible Hall-MHD System with Initial Magnetic Field Belonging to \(H^{\frac{3}{2}}({\mathbb {R}}^3)\). J. Math. Fluid Mech. 25, 20 (2023). https://doi.org/10.1007/s00021-023-00766-y
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DOI: https://doi.org/10.1007/s00021-023-00766-y