Global Strong Solutions to Density-Dependent Viscosity Navier-Stokes Equations in 3D Exterior Domains

TL;DR

This paper proves the global existence and exponential decay of strong solutions to 3D density-dependent Navier-Stokes equations in exterior domains, allowing vacuum states and small initial velocity gradients.

Contribution

It establishes the first global strong solution results for 3D exterior domains with density-dependent viscosity and vacuum, including decay rates.

Findings

Global strong solutions exist under small initial velocity gradients.

Solutions exhibit exponential decay over time.

Vacuum states in initial density are permitted.

Abstract

The nonhomogeneous Navier-Stokes equations with density-dependent viscosity is studied in three-dimensional (3D) exterior domains with nonslip or slip boundary conditions. We prove that the strong solutions exists globally in time provided that the gradient of the initial velocity is suitably small. Here the initial density is allowed to contain vacuum states. Moreover, after developing some new techniques and methods, the large-time behavior of the strong solutions with exponential decay-in-time rates is also obtained.