Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$ $(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying $$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$ To accomplish this goal, we require a norm inflation coming from the coupling term $L(a)\unicode[STIX]{x1D6E5}u$ instead of $u\cdot \unicode[STIX]{x1D6FB}u$ and construct a new decomposition of the density.

Chen等人已经证明了可压缩的Navier–Stokes方程的不适定性。 [关于临界Besov空间中可压缩Navier–Stokes方程的不适定性，Revista Mat。伊比利亚 31（2015），1375–1402]在临界Besov空间 $ L ^ {p} $ $（p> 6）$ 框架中。在本文中，我们用满足 $$ \ begin {eqnarray} \ displaystyle \ Vert \ unicode [STIX] {x1D70C} _ {0}-\ bar {\ unicode [STIX] {x1D70C}}的初始数据证明不适定性\ Vert _ {{\\ dot {B}} _ {p，1} ^ {\ frac {3} {p}}} \ leqslant \ unicode [STIX] {x1D6FF}，\ quad \ Vert u_ {0} \ Vert _ {{\\ dot {B}} _ {6,1} ^ {-\ frac {1} {2}}} \ leqslant \ unicode [STIX] {x1D6FF}。＆＆\ displaystyle \ nonumber \ end {eqnarray} $$ 为了实现这个目标，我们需要一个来自耦合项 $ L（a）\ unicode [STIX] {x1D6E5} u $ 而不是 $ u \ cdot \ unicode [STIX] {x1D6FB} u $的范式膨胀， 并构造一个新的密度分解。