Optimal well-posedness for the inhomogeneous incompressible Navier-Stokes system with general viscosity

In this paper we obtain new well-possedness results concerning a linear inhomogenous Stokes-like system. These results are used to establish local well-posedness in the critical spaces for initial density $\rho_{0}$ and velocity $u_{0}$ such that $\rho_{0}-\rho\in\dot{B}_{p,1}^{\frac{3}{p}}(\mathbb{R}^{3})$, $u_{0}\in\dot{B}_{p,1}^{\frac{3}{p}-1}(\mathbb{R}^{3})$, $p\in\left( \frac{6}{5},4\right) $, for the inhomogeneous incompressible Navier-Stokes system with variable viscosity. To the best of our knowledge, regarding the $3D$ case, this is the first result in a truly critical framework for which one does not assume any smallness condition on the density.