Strong Ill-Posedness in [math] of the 2D Boussinesq Equations in Vorticity Form and Application to the 3D Axisymmetric Euler Equations

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5915-5968, October 2024.
Abstract. We prove the strong ill-posedness of the two-dimensional Boussinesq system in vorticity form in [math] without boundary, building upon the method that Elgindi and Shikh Khalil [Strong Ill-Posedness in [math] for the Riesz Transform Problem, arXiv:2207.04556v1, 2022] developed for scalar equations. We provide examples of initial data with vorticity and density gradient of small [math] size, for which the horizontal density gradient [math] has a strong [math]-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 [math] provides a solution whose gradient of the swirl has a strong [math]-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.