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 with initial data $(u_0,B_0)in H^{frac{1}{2}+sigma}(mathbb{R}^3)times H^{frac{3}{2}}(mathbb{R}^3)$ and $sigmain (0,2)$. 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 cite{MR4193644}. Moreover, we establish the global well-posedness in $H^{frac{1}{2}+sigma}(mathbb{R}^3)times H^{frac{3}{2}}(mathbb{R}^3)$ with $sigmain (0,2)$ for small initial data and get the optimal time-decay rates of solutions. Our results improves some previous works.