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

Abstract

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.

Many typos corrected; a few references added. In the current version, we added estimates of several terms in technical lemmas that were missing