Strong illposedness of the incompressible Euler equation in integer C^m spaces

Abstract

We consider the d-dimensional incompressible Euler equations. We show strong illposedness of velocity in any Cm spaces whenever m ≥ 1 is an integer. More precisely, we show for a set of initial data dense in the Cm topology, the corresponding solutions lose Cm regularity instantaneously in time. In the C1 case, our proof is based on an anisotropic Lagrangian deformation and a short-time flow expansion. In the Cm, m ≥ 2 case, we introduce a flow decoupling method which allows to tame the nonlinear flow almost as a passive transport. The proofs also cover illposedness in Lipschitz spaces Cm−1,1 whenever m ≥ 1 is an integer.