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On the convergence rates for the compressible Navier-Stokes equations with potential force. (English) Zbl 1314.35085
Summary: In this paper, we are concerned with the optimal convergence rates of the global strong solution to the stationary solutions for the compressible Navier-Stokes equations with a potential external force \(\nabla \Phi\) in the whole space \(\mathbb R^{n}\) for \(n \geq 3\). It is proved that the perturbation and its first-order derivatives decay in \(L^{2}\) norm in \(O(t^{-n/4})\) and \(O(t^{-n/4-1/2})\), respectively, which are of the same order as those of the \(n\)-dimensional heat kernel, if the initial perturbation is small in \(H^{s_{0}}(\mathbb R^{n}) \cap L^{1}(\mathbb{R}^{n})\) with \(s_{0}=[n/2]+ 1\) and the potential force \(\Phi\) is small in some Sobolev space. The results also hold for \(n \geq 2\) when \(\Phi = 0\). When \(\Phi = 0\), we also obtain the decay rates of higher-order derivatives of perturbations.

MSC:
35Q30 Navier-Stokes equations
76N15 Gas dynamics, general
35D35 Strong solutions to PDEs
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