Norm-inflation with Infinite Loss of Regularity for Periodic NLS Equations in Negative Sobolev Spaces

In this paper we consider Schrodinger equations with nonlinearities of odd order 2σ + 1 on Td. We prove that for σd ≥2, they are strongly illposed in the Sobolev space Hs for any s < 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schrödinger equation and its renormalized version we prove such a result for Hs with s < −2/3.