Norm inflation for the derivative nonlinear Schrödinger equation

In this note, we study the ill-posedness problem for the derivative nonlinear Schr & ouml;dinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity for the gauged DNLS. This ill-posedness result is sharp since DNLS is known to be globally well-posed in L 2 (R) [16]. The main novelty of our approach is to control the derivative loss from the cubic nonlinearity by the quintic nonlinearity with carefully chosen initial data.