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In this paper, under the hypothesis that $\rho$ is upper bounded, weconstruct a Lyapunov functional for the multidimensional isentropiccompressible Navier-Stokes equations and show that the weak solutions decayexponentially to the equilibrium state in $L^2$ norm. This can be regarded as ageneralization of Matsumura and Nishida's results in 1982, since our analysisis done in the framework of Lions 1998 and Feireisl et al. 2001, the higherregularity of $(\rho, u)$ and the uniformly positive lower bound of $\rho$ arenot necessary in our analysis and vacuum may be admitted. Indeed, the upperbound of the density $\rho$ plays the essential role in our proof.

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