Global well-posedness of the incompressible Hall-MHD system in critical spaces

In this paper, we consider the initial value problem of the incompressible Hall-MHD system and prove the global well-posedness in the scaling critical class \({\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}(\mathbb {R}^3)\times ({\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}(\mathbb {R}^3) \cap L^{\infty }(\mathbb {R}^3))\) ${\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}({\mathbb{R}}^3)\times ({\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}({\mathbb{R}}^3)\cap L^{\infty }({\mathbb{R}}^3))$ for \(3< p < \infty \) $3<p<\infty$ . Moreover, we also refine the smallness conditions and show that our global well-posedness holds for initial data whose \({\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}(\mathbb {R}^3)\) ${\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}({\mathbb{R}}^3)$ -norm is large, provided that some weaker norm is sufficiently small.