Ill-posedness for the incompressible Euler equations in critical Sobolev spaces

Tarek Mohamed Elgindi, In-Jee Jeong

Published 2016-03-25Version 1

For the $2D$ Euler equation on the torus with vorticity formulation, we construct localized smooth solutions whose $H^1$-norm becomes large in a short period of time. This gives a simple alternative proof of the recent ill-posedness result of Bourgain-Li \cite{MR3359050}. Our main observation is that a localized chunk of vorticity with odd symmetry is able to generate a hyperbolic flow at least for a small time, which stretches the vorticity gradient.