Well-posedness for the incompressible Hall-MHD system with initial magnetic field belonging to \(H^{\frac{3}{2}}(\mathbb{R}^3)\)

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.