Optimal decay of compressible Navier-Stokes equations with or without potential force

Abstract

In this paper, we investigate the optimal decay rate for the higher order spatial derivative of global solution to the compressible Navier-Stokes (CNS) equations with or without potential force in three-dimensional whole space. First of all, we show that the N-th order spatial derivative of global small solution of the CNS equations without potential force tends to zero at the rate (1+t)−(s+N) instead of (1+t)−(s+N−1) stated in [13]. Secondly, we study optimal decay rate for global solution of the CNS equations with potential force. The tricky problem comes from the equilibrium state that will depend on spatial variable rather than constant. Based on energy estimate, spectral analysis, and high-low frequency decomposition, we establish the upper and lower bounds of decay rates for the solution itself and its spatial derivatives. These decay rates are really optimal since the upper bounds of decay rates coincide with the lower ones.