Optimal decay of full compressible Navier-Stokes equations with potential force

Abstract

In this paper, we aim to investigate the optimal decay rate for the higher order spatial derivative of global solution to the full compressible Navier-Stokes (CNS) equations with potential force in $\mathbb{R}^3$. We establish the optimal decay rate of the solution itself and its spatial derivatives (including the highest order spatial derivative) for global small solution of the full CNS equations with potential force. With the presence of potential force in the considered full CNS equations, the difficulty in the analysis comes from the appearance of non-trivial ststionary solutions. These decay rates are really optimal in the sense that it coincides with the rate of the solution of the linerized system. In addition, the proof is accomplished by virtue of time weighted energy estimate, spectral analysis, and high-low frequency decomposition.