Global stability and non-vanishing vacuum states of 3D compressible Navier-Stokes equations

We investigate global stability and non-vanishing vacuum states of large solutions to the compressible Navier-Stokes equations on the torus , and the main purpose of this work is three-fold: 

(1) Under the assumption that the density ρ(x,t) verifies sup_{t\geq 0} ||ρ(x,t)||_{L^{\infty}}<M,   it is shown that the solutions converge to equilibrium state exponentially in -norm. In contrast to the previous related works where the density has uniform positive lower and upper bounds, this gives the first stability result for large strong solutions of the 3D compressible Navier-Stokes equations in the presence of vacuum. 

(2) By employing some new thoughts, we also show that the density converges to its equilibrium state exponentially in L^{\infty}-norm if additionally the initial density ρ_0(x) satisfies 

 inf_{x \in T^3}||ρ_0(\cdot)||_{L^{\infty}} \geq c_0>0 . 

(3) We prove that the vacuum state will persist for any time provided that the initial density contains vacuum, which is different from the previous work of [H. L. Li et al., Commun. Math. Phys., 281 (2008), 401-444], where the authors  showed that any vacuum state must vanish within finite time for the free boundary problem of  the 1D compressible Navier-Stokes equations with density-dependent viscosity μ(ρ)=ρ^{α} with α>1/2.