Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density

TL;DR

This paper proves the global existence and uniqueness of small solutions to the inhomogeneous Navier-Stokes equations in the half-space with bounded initial density close to a positive constant, extending some results to bounded domains.

Contribution

It establishes global existence and uniqueness results for inhomogeneous Navier-Stokes equations with minimal density regularity and extends findings to bounded domains.

Findings

Global existence of small solutions in half-space

Uniqueness with additional initial velocity regularity

Partial extension to bounded domains

Abstract

In this paper, we establish the global existence of small solutions to the inhomogeneous Navier-Stokes system in the half-space. The initial density only has to be bounded and close enough to a positive constant, and the initial velocity belongs to some critical Besov space. With a little bit more regularity for the initial velocity, those solutions are proved to be unique. In the last section of the paper, our results are partially extended to the bounded domain case.