Global solvability of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity

TL;DR

This paper proves the global existence and uniqueness of strong solutions for 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, even with vacuum and large initial density, extending previous results.

Contribution

It establishes the global solvability of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity under new conditions, including vacuum and large initial density.

Findings

Global-in-time unique strong solutions exist under small initial velocity gradient.

Results hold even with vacuum and large initial density.

Generalizes previous results for constant viscosity.

Abstract

In this paper, we consider the three-dimensional inhomogeneous Navier-Stokes equations with density-dependent viscosity in presence of vacuum over bounded domains. Global-in-time unique strong solution is proved to exist when $\|\nabla u_0\|_{L^2}$ is suitably small with arbitrary large initial density. This generalizes all the previous results even for the constant viscosity.