Global Wellposeness for the 3D inhomogeneous incompressible Navier-Stokes equations

TL;DR

This paper proves the global existence and uniqueness of strong solutions for 3D inhomogeneous incompressible Navier-Stokes equations with small initial data, allowing for initial densities that are not strictly positive.

Contribution

It establishes a new global well-posedness result for inhomogeneous Navier-Stokes equations with minimal initial density restrictions.

Findings

Global existence and uniqueness of strong solutions.

Initial density need not be strictly positive.

Results hold for small initial $ ext{dot-H}^{1/2}$-norm data.

Abstract

This paper addresses the three-dimensional Navier-Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data with small $\dot{H}^{\frac12}$-norm, which also satisfies a natural compatibility condition. A key point of the theorem is that the initial density need not be strictly positive.