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 a certain subspace of $\mathrm{BMO}^{-1}$, 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.