Strong ill-posedness in W1,? of the 2d stably stratified Boussinesq equations and application to the 3d axisymmetric Euler Equations.

We prove the strong ill-posedness of the two-dimensional Boussinesq system in vorticity form in L8pR2q without boundary, building upon the method that Shikh Khalil & Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide examples of initial data with vorticity and density gradient of small L8pR2q size, for which the horizontal density gradient has a strong L8pR2q-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 our 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 vorticity uniformly bounded and small in L8pR2q provides a solution whose gradient of the swirl has a strong L8pR2q-norm inflation in infinitesimal time. The norm inflations are quantified from below by an explicit lower bound which depends on time, the size of the data and is valid for small times