Ill-posedness for the Incompressible Euler Equations in Critical Sobolev Spaces

For the 2D Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $$L^\infty \cap H^1$$ but escapes $$H^1$$ immediately for $$t>0$$ . Our main observation is that a localized chunk of vorticity bounded in $$L^\infty \cap H^1$$ with odd-odd symmetry is able to generate a hyperbolic flow with large velocity gradient at least for a short period of time, which stretches the vorticity gradient.