  The present paper is dedicated to the global well-posedness for the 3D
inhomogeneous incompressible Navier-Stokes equations, in critical Besov spaces
without smallness assumption on the variation of the density. We aim at
extending the work by Abidi, Gui and Zhang (Arch. Ration. Mech. Anal. 204
(1):189--230, 2012, and J. Math. Pures Appl. 100 (1):166--203, 2013) to a more
lower regularity index about the initial velocity. The key to that improvement
is a new a priori estimate for an elliptic equation with nonconstant
coefficients in Besov spaces which have the same degree as $L^2$ in
$\mathbb{R}^3$. Finally, we also generalize our well-posedness result to the
inhomogeneous incompressible MHD equations.
