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Large time behavior of isentropic compressible Navier-Stokes system in $$\mathbb R^3$$. (English) Zbl 1214.35047
Summary: We consider the long-time behavior and optimal decay rates of global strong solution to three-dimensional isentropic compressible Navier-Stokes (CNS) system in the present paper. When the regular initial data also belong to some Sobolev space $$H^l(\mathbb R^3)\cap\dot B_{1,\infty}^{-s}(\mathbb R^3)$$ with $$l\geq 4$$ and $$s\in [0,1]$$, we show that the global solution to the CNS system converges to the equilibrium state at a faster decay rate in time. In particular, the density and momentum converge to the equilibrium state at the rates $$(1+t)^{-3/4-s/2}$$ in the $$L^2$$-norm or $$(1+t)^{-3/2-s/2}$$ in the $$L^\infty$$-norm, respectively, which are shown to be optimal for the CNS system.

##### MSC:
 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35B40 Asymptotic behavior of solutions to PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
##### Keywords:
compressible Navier-Stokes system; optimal decay rate
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##### References:
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