Density-dependent incompressible Navier--Stokes equations in critical tent spaces

In this article, we prove the existence of global solutions to the inhomogeneous incompressible Navier–Stokes equations, whenever the initial velocity belongs to some subspace of ${\mathrm{B}\mathrm{M}\mathrm{O}}^{-1}$, and the initial density is sufficiently close to $1$ in the uniform metric. This is a natural extension to the variable density case of the celebrated result by H. Koch and D. Tataru concerning the classical Navier-Stokes equations.