Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MUD equations

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 (2012) [2], and (2013) [3] to a 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 R-3. Finally, we also generalize our well-posefiness result to the inhomogeneous incompressible MHD equations. (C) 2016 Elsevier Inc. All rights reserved.