Well-Posedness for the Incompressible Hall-MHD System with Initial Magnetic Field Belonging to $$H^{\frac{3}{2}}({\mathbb {R}}^3)$$

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.