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 Nishidaʼs results [A. Matsumura, T. Nishida, in: Computing Methods in Applied Sciences and Engineering, vol. V, 1982, pp. 389–406], since our analysis is done in the framework of Lions [P.-L. Lions, Oxford Science Publication, 1998] and Feireisl et al. [E. Fereisl, A. Novotny, H. Petzeltová, J. Math. Fluid Mech. 3 (2001) 358–392], 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.