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

Given initial data (ρ0, u0) satisfying 0 < m ⩽ ρ0 ⩽ M, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0 - 1 \in L^2 \cap \dot W^{1,r} (R^3 )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0 \in \dot H^{ - 2\delta } \cap H^1 (\mathbb{R}^3 )$$\end{document} for δ ∈]1/4, 1/2[ and r ∈]6, 3/1 − 2δ[, we prove that: there exists a small positive constant ɛ1, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left\| {u_0 } \right\|_{L^2 } \left\| {\nabla u_0 } \right\|_{L^2 } $\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left\| {\mu (\rho _0 ) - 1} \right\|_{L^\infty } \leqslant \varepsilon _0 $\end{document} for some uniform small constant ɛ0. Furthermore, with smoother initial data and viscosity coefficient, we can prove the propagation of the regularities for such strong solution.