PUB - Publikationen an der Universität Bielefeld

We prove the strong ill-posedness of the two-dimensional Boussinesq system in vorfor scalar equations. We provide examples of initial data with vorticity and density gradient of small LOO(R2) size, for which the horizontal density gradient \partial x\rho has a strong LOO(R2)-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 LOO(R2) provides a solution whose gradient of the swirl has a strong LOO(R2)-norm inflation in infinitesimal time. The norm inflation is quantified from below by an explicit lower bound which depends on time and the size of the data and is valid for small times.