For quadratic derivative fractional NLS, the Cauchy problem is ill-posed in Sobolev spaces below sharp fractional derivative exponents due to norm inflation with infinite loss of regularity.
Chihara,The initial value problem for Schr¨ odinger equations on the torus, Int
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Norm inflation for quadratic derivative fractional nonlinear Schr\"odinger equations
For quadratic derivative fractional NLS, the Cauchy problem is ill-posed in Sobolev spaces below sharp fractional derivative exponents due to norm inflation with infinite loss of regularity.