Norm inflation for quadratic derivative fractional nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem for quadratic derivative fractional nonlinear Schrödinger equations on $\mathbb{R}$ or $\mathbb{T}$. We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed in the Sobolev space. Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms. By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.