On the Cauchy Problem of 3D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

TL;DR

This paper proves the global existence and decay of strong solutions to 3D nonhomogeneous Navier-Stokes equations with vacuum and density-dependent viscosity, without smallness conditions on initial density.

Contribution

It establishes the global existence of strong solutions with vacuum and density-dependent viscosity in 3D, using new a priori decay estimates and allowing initial vacuum states.

Findings

Proves exponential decay rates of solutions over time.

Shows global existence without small initial density assumptions.

Allows initial vacuum and compact support in density.

Abstract

We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity and vacuum. We establish some key a priori exponential decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence of strong solutions in the whole three-dimensional space, provided that the initial velocity is suitably small in the $\dot H^\beta$-norm for some $\beta\in(1/2,1].$ Note that this result is proved without any smallness conditions on the initial density. Moreover, the density can contain vacuum states and even have compact support initially.