On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity

Given solenoidal vector $u_0\in H^{-2\text{d}}\cap H^1({\mathbb{R}}^2),$ $\r_0-1\in L^2(\R^2),$and$\r_0 \in L^\infty\cap\dot{W}^{1,r}(\R^2)$ with a positive lower bound for $\text{d}\in (0,\text{f}12)$ and $2<r<\text{f}21-2\text{d},$ we prove that 2-D incompressible inhomogeneous Navier-Stokes system \eqref{1.1} has a unique global solution provided that the viscous coefficient \mu(\r_0) is close enough to 1 in the $L^{\mathrm{\infty }}$ norm compared to the size of $\text{d}$ and the norms of the initial data. With smoother initial data, we can prove the propagation of regularities for such solutions. Furthermore, for $1<p<4,$ if (\r_0-1,u_0) belongs to the critical Besov spaces $\dB^{\f2p}{p,1}(\R^2)\times \bigl(\dB^{-1+\f2p}{p,1}\cap L^2(\R^2)\bigr)$andthe$\dB^{\f2p}{p,1}(\R^2)$normof$\r_0-1$issufficientlysmallcomparedtotheexponentialof$|u_0|{L^2}^2+|u_0|{\dB^{-1+\f2p}{p,1}},$ we prove the global well-posedness of \eqref{1.1} in the scaling invariant spaces. Finally for initial data in the almost critical Besov spaces, we prove the global well-posedness of \eqref{1.1} under the assumption that the $L^{\mathrm{\infty }}$ norm of \r_0-1 is sufficiently small.