# Optimal decay for the full compressible Navier-Stokes system in critical $L^p$ Besov spaces

## TL;DR

This paper establishes optimal decay rates for solutions to the full compressible Navier-Stokes system in critical Besov spaces, extending previous results by removing smallness assumptions on initial data at low frequencies.

## Contribution

It proves optimal decay rates for solutions in critical Besov spaces without requiring small initial data at low frequencies, using a pure energy method.

## Key findings

- Solutions decay at the optimal rate $(1+t)^{-rac{N}{2}(rac{1}{2}-rac{1}{p})-rac{s+\sigma_1}{2}}$.
- The decay results hold under broader initial data conditions, removing smallness constraints.
- The approach avoids spectral analysis, simplifying the proof of decay in critical spaces.

## Abstract

Danchin and He (Math. Ann. 64: 1-38, 2016) recently established the global existence in critical $L^p$-type regularity framework for the $N$-dimensional $(N\geq 3)$ non-isentropic compressible Navier-Stokes equations. The purpose of this paper is to further investigate the large time behavior of solutions constructed by them. More precisely, we prove that if the initial data at the low frequencies additionally belong to some Besov space $\dot{B}_{2,\infty}^{-\sigma_1}$ with $\sigma_1\in (2-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^s$ norm of the critical global solutions exhibits the optimal decay $(1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{s+\sigma_1}{2}}$ for suitable $p$ and $s$. The main tool we use is the pure energy argument without the spectral analysis, which enables us to \emph{remove the smallness assumption} of initial data at the low-frequency.