Global well-posedness of 3D inhomogenous incompressible Navier-Stokes equations with density-dependent viscosity

Abstract

The issue of global well-posedness for the 3D inhomogenous incompressible Navier-Stokes equations was first addressed by Kazhikov in 1974. In this manuscript, we obtain its global well-posedness for the system with density-dependent viscosity under the smallness assumption of initial velocity in the critical space $\dot{B}_{p,1}^{-1+\frac 3p}$ with $p\in ]1, 9/2]$. To the best of our knowledge, this is the first result about the global well-posedness for which one does not assume any smallness condition on the density when the initial density is far away from vacuum.