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

TL;DR

This paper determines the optimal decay rates for solutions and their derivatives in the full compressible Navier-Stokes equations with potential force, using advanced analytical techniques to match linearized system decay.

Contribution

It establishes the optimal decay rates for solutions and derivatives of the full CNS equations with potential force, including the highest order derivatives, using spectral and energy methods.

Findings

Decay rates match those of the linearized system.

Optimal decay rates for all derivatives including highest order.

Analysis handles stationary solutions caused by 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.