Global unique solutions for the inhomogeneous Navier-Stokes equation with only bounded density, in critical regularity spaces

TL;DR

This paper proves the global existence and uniqueness of solutions to the inhomogeneous Navier-Stokes equations with bounded, possibly discontinuous density, in critical regularity spaces, extending previous results to cases with large density variations and vacuum.

Contribution

It establishes global well-posedness for the inhomogeneous Navier-Stokes system with discontinuous density in critical spaces, including large density variations and vacuum cases.

Findings

Global existence and uniqueness in 2D for initial velocity in critical Besov spaces.

Extension of uniqueness to large density variations and vacuum.

Application of Lorentz space estimates and interpolation techniques.

Abstract

We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where the initial density is discontinuous and the initial velocity has critical regularity. Assuming that the initial density is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity belongs to some critical homogeneous Besov space (and in small in the three-dimensional case). Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the case of large variations of density with, possibly, vacuum. Interestingly, our result implies that the Fujita-Kato type solutions constructed by P. Zhang in are unique. Our work relies on interpolation results, time weighted estimates and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system.