Global strong solution of 3D inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity

Abstract

In this paper, we investigate the Cauchy problem of the three-dimensional inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity. For the case μ(ρ)=ρ, we prove the global existence of strong solution for small initial data satisfying the compatibility condition. Furthermore, we also obtain the large time algebraic decay rates of the velocity. The innovation of this article is that it does not require a positive lower bound for viscosity.