Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces

For the -dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space , . The borderline case was a folklore open problem. In this paper we consider the physical dimension and show that if we perturb any given smooth initial data in norm, then the corresponding solution can have infinite norm instantaneously at . In a companion paper [1] we settle the 3D and more general cases. The constructed solutions are unique and even -smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems.