  We prove the strong ill-posedness in the sense of Hadamard of the
two-dimensional Boussinesq equations in $W^{1, \infty}(\mathbb{R}^2)$ without
boundary, extending to the case of systems the method that Shikh Khalil \&
Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide a large
class of initial data with velocity and density of small $W^{1,
\infty}(\mathbb{R}^2)$ size, for which the horizontal density gradient has a
strong $L^{\infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, while
the vorticity and the vertical density gradient remain bounded. Furthermore,
exploiting the three-dimensional version of Elgindi's decomposition of the
Biot-Savart law, we apply the method to the three-dimensional axisymmetric
Euler equations with swirl and away from the vertical axis, showing that a
large class of initial data with velocity field uniformly bounded in $W^{1,
\infty}(\mathbb{R}^2)$ provides a solution whose swirl component has a strong
$W^{1, \infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, while the
potential vorticity remains bounded at least for small times. Finally, the
$W^{1,\infty}$-norm inflation of the swirl (and the $L^{\infty}$-norm inflation
of the vorticity field) is quantified from below by an explicit lower bound
which depends on time, the size of the data and it is valid for small times.
