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Decay estimates for isentropic compressible Navier-Stokes equations in bounded domain. (English) Zbl 1229.35171
Summary: 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 A. Matsumura and T. Nishida’s results [Computing methods in applied sciences and engineering V, Proc. 5th int. Symp., Versailles 1981, 389–406 (1982; Zbl 0505.76083)], since our analysis is done in the framework of P.-L. Lions [Mathematical topics in fluid mechanics. Vol. 2: Compressible models. Oxford: Clarendon Press (1998; Zbl 0908.76004)] and E. Feireisl, A. Novotný and H. Petzeltová [J. Math. Fluid Mech. 3, No. 4, 358–392 (2001; Zbl 0997.35043)], 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.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35D30 Weak solutions to PDEs
##### Keywords:
compressible Navier-Stokes equations; decay estimates
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