Spectral numerical results
On the range $\sigma\in[0,500]$, there exists a unique axisymmetric self-similar solution $\bU_\sigma$ that is symmetric with respect to the plane $z=0$. The spectrum of the linearization $DF(\bU_\sigma)$ has the form $$\sigma(DF(\bU_\sigma))\subset\bigl\{\lambda\in\mathbb{C}\,:\:\mathrm{Re}\,\lambda>\delta\bigr\}\cup\bigl\{\lambda_{\sigma}\bigr\}\,,\tag{$*$}$$
for some $\delta>0$ and there exists $\sigma_{0}\approx292$ such that $\lambda_{\sigma}>0$ for $\sigma<\sigma_{0}$, $\lambda_{\sigma}=0$ for $\sigma=\sigma_{0}$, and $\lambda_{\sigma}<0$ for $\sigma>\sigma_{0}$.
Nonuniqueness numerical results
There exists two different smooth solutions in $L^\infty(0,T;L^{3,\infty}(\mathbb{R}^3))$ with the same scale-invariant initial data $\bu_0\in L^{3,\infty}(\mathbb{R}^3)$.