Norm inflation for the derivative nonlinear Schrödinger equation

In this note, we study the ill-posedness problem for the derivative nonlinear Schrödinger 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℝ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> [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.