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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
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