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

Abstract

For the d-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space (Formula presented.), (Formula presented.). The borderline case (Formula presented.) was a folklore open problem. In this paper we consider the physical dimension d=2 and show that if we perturb any given smooth initial data in (Formula presented.) norm, then the corresponding solution can have infinite (Formula presented.) norm instantaneously at (Formula presented.). In a companion paper [1] we settle the 3D and more general cases. The constructed solutions are unique and even (Formula presented.)-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.