Abstract
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
Anno
2023
Autori IAC
Tipo pubblicazione
Altri Autori
Roberta Bianchini, Lars Eric Hientzsch, Felice Iandoli
Editore
Cornell University
Rivista
arXiv.org