Norm inflation for quadratic derivative fractional nonlinear Schr\"odinger equations

TL;DR

This paper investigates the well-posedness of quadratic derivative fractional nonlinear Schrödinger equations, identifying critical fractional derivatives and demonstrating ill-posedness through norm inflation in Sobolev spaces.

Contribution

It determines the sharp fractional derivative exponents for well-posedness and proves norm inflation results, extending understanding of ill-posedness in these equations.

Findings

Identified sharp fractional derivative exponents for well-posedness.

Established norm inflation with infinite regularity loss.

Proved ill-posedness in certain Sobolev spaces.

Abstract

We consider the Cauchy problem for quadratic derivative fractional nonlinear Schr\"odinger 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.