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

Abstract

In this paper we obtain new well-posedness results concerning a linear inhomogeneous 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 ${\rho }_0-\rho \in {Ḃ}_{p,1}^{3∕p}\left({ℝ}^3\right)$, $u_0\in {Ḃ}_{p,1}^{3∕p-1}\left({ℝ}^3\right)$, $p\in \left(\frac{6}{5},4\right)$ for the inhomogeneous incompressible Navier–Stokes system with variable viscosity. To the best of our knowledge, regarding the 3-dimensional case, this is the first result in a truly critical framework for which one does not assume any smallness condition on the density.