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

Abstract

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.