On the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes system with density-dependent viscosity

In this paper, we are concerned with the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity when the initial velocity is sufficiently small in the critical Besov space ${\dot{B}}^{\frac{1}{2}}$. Compared with the previous result of Abidi and Zhang (Science China Mathematics 58 (6) (2015) 1129-1150), we remove the smallness assumption of the viscosity $\mu ({\rho }_0)-1$ in $L^{\mathrm{\infty }}$-norm.