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Decay Estimates for Isentropic Compressible Navier-Stokes Equations in Bounded Domain

In this paper, under the hypothesis that $ρ$ 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 $(ρ, u)$ and the uniformly positive lower bound of $ρ$ are not necessary in our analysis and vacuum may be admitted. Indeed, the upper bound of the density $ρ$ plays the essential role in our proof.