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We prove the strong ill-posedness in the sense of Hadamard of thetwo-dimensional Boussinesq equations in $W^{1, \infty}(\mathbb{R}^2)$ withoutboundary, extending to the case of systems the method that Shikh Khalil \&Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide a largeclass of initial data with velocity and density of small $W^{1,\infty}(\mathbb{R}^2)$ size, for which the horizontal density gradient has astrong $L^{\infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, whilethe vorticity and the vertical density gradient remain bounded. Furthermore,exploiting the three-dimensional version of Elgindi's decomposition of theBiot-Savart law, we apply the method to the three-dimensional axisymmetricEuler equations with swirl and away from the vertical axis, showing that alarge 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 thepotential vorticity remains bounded at least for small times. Finally, the$W^{1,\infty}$-norm inflation of the swirl (and the $L^{\infty}$-norm inflationof the vorticity field) is quantified from below by an explicit lower boundwhich depends on time, the size of the data and it is valid for small times.

Strong ill-posedness in $W^{1, \infty}$ of the 2d stably stratified Boussinesq equations and application to the 3d axisymmetric Euler Equations