On the well-posedness of 3-D inhomogeneous incompressible Navier-Stokes equations with variable viscosity

In this paper, we mainly study the well-posedness for the 3-D inhomogeneous incompressible Navier-Stokes equations with variable viscosity. With some smallness assumption on the BMO-norm of the initial density, we first get the local well-posedness of (1.1) in the critical Besov spaces. Moreover, if the viscosity coefficient is a constant, we can extend this local solution to be a global one. Our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by Abidi, Gui and Zhang in [3], Burtea in [8] and Zhai and Yin in [32] to approach the ideal one i.e. 1