  Singularity formation of the 3D incompressible Euler equations is known to be
extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler
equations with no swirl and $C^{1,\alpha}$ initial velocity develops a finite
time singularity. Inspired by Elgindi's work, we proved that the 3D
axisymmetric Euler and 2D Boussinesq equations with $C^{1,\alpha}$ initial
velocity and boundary develop a stable asymptotically (or approximately)
self-similar finite time singularity [8]. On the other hand, the authors of
[35,52] recently showed that blowup solutions to the 3D Euler equations are
hydrodynamically unstable. The instability results obtained in [35,52] require
some strong regularity assumption on the initial data, which is not satisfied
by the $C^{1,\alpha}$ velocity field. In this paper, we generalize the analysis
of [8,18,35,52] to show that the blowup solutions of the 3D Euler and 2D
Boussinesq equations with $C^{1,\alpha}$ velocity are unstable under the notion
of stability introduced in [35,52]. These two seemingly contradictory results
reflect the difference of the two approaches in studying the stability of 3D
Euler blowup solutions. The stability analysis of the blowup solution obtained
in [8,18] is based on the stability of a dynamically rescaled blowup profile in
space and time, which is nonlinear in nature. The linear stability analysis in
[35,52] is performed by directly linearizing the 3D Euler equations around a
blowup solution in the original variables. It does not take into account the
changes in the blowup time, the dynamic changes of the rescaling rate of the
perturbed blowup profile and the blowup exponent of the original 3D Euler
equations using a perturbed initial condition when there is an approximate
self-similar blowup profile. Such information has been used in an essential way
in establishing the nonlinear stability of the blowup profile in [8,18,19].
