Global well-posedness of 3-D density-dependent Navier-Stokes system with variable viscosity

Given initial data (ρ ₀, u ₀) satisfying 0 < m ? ρ ₀ ? M, (rho _0 - 1 in L^2 cap dot W^{1,r} (R^3 )) and (u_0 in dot H^{ - 2delta } cap H^1 (mathbb{R}^3 )) for δ ∈]1/4, 1/2[ and r ∈]6, 3/1 ? 2δ[, we prove that: there exists a small positive constant ? ₁, which depends on the norm of the initial data, so that the 3-D incompressible inhomogeneous Navier-Stokes system with variable viscosity has a unique global strong solution (ρ, u) whenever (left| {u_0 } right|_{L^2 } left| {nabla u_0 } right|_{L^2 } ) and (left| {mu (rho _0 ) - 1} right|_{L^infty } leqslant varepsilon _0 ) for some uniform small constant ? ₀. Furthermore, with smoother initial data and viscosity coefficient, we can prove the propagation of the regularities for such strong solution.