We consider the d-dimensional incompressible Euler equations. We show
strong illposedness of velocity in any Cm spaces whenever mβ₯1 is an
\emph{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, mβ₯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 Cmβ1,1 whenever mβ₯1 is an integer.Comment: 76 pages. Minor corrections. To appear in GAF