  It is proved in \cite{IO21} that the Cauchy problem for the full compressible
Navier--Stokes equations of the ideal gas is ill-posed in $\dot{B}_{p, q}^{2 /
p}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-1}(\mathbb{R}^2) \times
\dot{B}_{p, q}^{2 / p-2}(\mathbb{R}^2) $ with $1\leq p\leq \infty$ and $1\leq
q<\infty$. In this paper, we aim to solve the end-point case left in
\cite{IO21} and prove that the Cauchy problem is ill-posed in $\dot{B}_{p,
\infty}^{d / p}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d /
p-1}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-2}(\mathbb{R}^d)$ with
$1\leq p\leq\infty$ by constructing a sequence of initial data which shows that
the solution map is discontinuous at zero. As a by-product, we demonstrate that
the incompressible Navier--Stokes equations is also ill-posed in
$\dot{B}_{p,\infty}^{d/p-1}(\mathbb{R}^d)$, which is an interesting open
problem in itself.
