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 L8pR2qwithout boundary, building upon the method that Shikh Khalil & Elgindi arXiv:2207.04556v1 developed for scalarequations. We provide examples of initial data with vorticity and density gradient of small L8pR2q size, for which thehorizontal density gradient has a strong L8pR2q-norm inflation in infinitesimal time, while the vorticity and the verticaldensity gradient remain bounded. Furthermore, exploiting the three-dimensional version of Elgindi's decomposition ofthe Biot-Savart law, we apply our method to the three-dimensional axisymmetric Euler equations with swirl and awayfrom the vertical axis, showing that a large class of initial data with vorticity uniformly bounded and small in L8pR2qprovides a solution whose gradient of the swirl has a strong L8pR2q-norm inflation in infinitesimal time. The norminflations are quantified from below by an explicit lower bound which depends on time, the size of the data and is validfor small times