On the convergence rates for the compressible Navier-Stokes equations with potential force

Abstract

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 ∇Φ in the whole space ℝn for n ≥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 s0(ℝn) ∩L 1 (ℝn) with s0 =[n/2]+ 1 and the potential force Φ is small in some Sobolev space. The results also hold for n ≥ 2 when Φ = 0. When Φ = 0, we also obtain the decay rates of higher-order derivatives of perturbations.