The $L^{p}$ theory for non-isentropic Navier-Stokes equations governing compressible viscous and heat-conductive gases is not yet proved completely so far, because the critical regularity cannot control all non linear coupling terms. In this paper, we pose an additional regularity assumption of low frequencies in $\mathbb{R}^d(d\geq 3)$, and then the sharp time-weighted inequality can be established, which leads to the time-decay estimates of global strong solutions in the $L^{p}$ critical Besov spaces. Precisely, we show that if the initial data belong to some Besov space $\dot{B}^{-s_{1}}_{2,\infty}$ with $s_{1}\in (1-\frac{d}{2}, s_0](s_0\triangleq \frac{2d}{p}-\frac{d}{2})$, then the $L^{p}$ norm of the critical global solutions admits the time decay $t^{-\frac{s_{1}}{2}-\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}$ (in particular, $t^{-\frac{d}{2p}}$ if $s_1=s_0$), which coincides with that of heat kernel in the $L^p$ framework. In comparison with \cite{DX2}, the low-frequency regularity $s_1$ can be improved to be \textit{the whole range}.

中文翻译：

---

可压缩粘性和导热气流的临界 Lp 框架中解的长时间行为

控制可压缩粘性和导热气体的非等熵 Navier-Stokes 方程的 $L^{p}$ 理论目前尚未完全证明，因为临界规律无法控制所有非线性耦合项。在本文中，我们在 $\mathbb{R}^d(d\geq 3)$ 中提出了一个额外的低频正则假设，然后可以建立尖锐的时间加权不等式，这导致时间衰减估计$L^{p}$ 临界 Besov 空间中的全局强解。准确地说，我们证明如果初始数据属于某个 Besov 空间 $\dot{B}^{-s_{1}}_{2,\infty}$ with $s_{1}\in (1-\frac{ d}{2}, s_0](s_0\triangleq \frac{2d}{p}-\frac{d}{2})$，那么临界全局解的 $L^{p}$ 范数承认时间衰减 $t^{-\frac{s_{1}}{2}-\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}$（特别是, $t^{-\frac{d}{2p}}$ if $s_1=s_0$)，这与 $L^p$ 框架中的热核一致。与 \cite{DX2} 相比，低频规律 $s_1$ 可以改进为 \textit{整个范围}。