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

TL;DR

This paper proves the global existence of solutions to the inhomogeneous incompressible Navier--Stokes equations with initial data in specific function spaces, extending classical results to variable density scenarios.

Contribution

It extends the classical Navier-Stokes existence results to the variable density case within critical tent spaces, under specific initial conditions.

Findings

Global solutions exist for initial velocities in certain BMO^{-1} subspaces.

Initial density close to 1 in the uniform metric ensures solution existence.

Extends classical results to inhomogeneous, variable density Navier-Stokes equations.

Abstract

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 $\mathrm{BMO}^{-1}$, 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.