  This paper presents some progress toward an open question which proposed by
Tsurumi (Arch. Ration. Mech. Anal. 234:2, 2019): whether or not the stationary
Navier-Stokes equations in $\R^d$ is well-posed from $\dot{B}_{p, q}^{-2}$ to
$\mathbb{P} \dot{B}_{p, q}^{0}$ with $p=d$ and $1 \leq q \leq 2$. In this
paper, we prove that for the case $1\leq q<\frac d2$ with $d\geq4$ the
stationary Navier-Stokes equations is ill-posed from $\dot{B}_{d,
q}^{-2}(\R^d)$ to $\mathbb{P} \dot{B}_{d, q}^{0}(\R^d)$ by showing that a
sequence of external forces is constructed to show discontinuity of the
solution map at zero. Indeed in such case of $q$, there exists a sequence of
external forces which converges to zero in $\dot{B}_{d, q}^{-2}$ and yields a
sequence of solutions which does not converge to zero in $\dot{B}_{d, q}^{0}$.
In particular, we also prove that the stationary Navier-Stokes equations is
well-posed from $\dot{B}_{d, 2}^{-2}(\R^d)$ to $\mathbb{P} \dot{B}_{d,
2}^{0}(\R^d)$ with $d=3,4$. Based on these two cases, we demonstrate that the
above open question for the dimension $d\geq4$ has been solved completely.
