Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VM{O}^{-1}$

In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $({\rho }_0,u_0)$ lying in $C^1\times (L^2\cap VM{O}^{-1})$, where ${\rho }_0$ has a positive lower bound. Furthermore, if ${\rho }_0\in C^2$ and $\mathrm{\mid }\mathrm{\mid }{\rho }_0-1\mathrm{\mid }{\mathrm{\mid }}_{L^{\mathrm{\infty }}}+\mathrm{\mid }\mathrm{\mid }{u}_0\mathrm{\mid }{\mathrm{\mid }}_{BM{O}^{-1}}$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.