Decay Estimates for Isentropic Compressible Navier-Stokes Equations in   Bounded Domain
  In this paper, under the hypothesis that $\rho$ is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible Navier-Stokes equations and show that the weak solutions decay exponentially to the equilibrium state in $L^2$ norm. This can be regarded as a generalization of Matsumura and Nishida's results in 1982, since our analysis is done in the framework of Lions 1998 and Feireisl et al. 2001, the higher regularity of $(\rho, u)$ and the uniformly positive lower bound of $\rho$ are not necessary in our analysis and vacuum may be admitted. Indeed, the upper bound of the density $\rho$ plays the essential role in our proof. 
Daoyuan Fang, Ruizhao Zi, Ting Zhang
math.AP