# zbMATH — the first resource for mathematics

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
##### Keywords:
compressible Navier-Stokes equations; convergence rate
Full Text: