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In this note, we consider the ill-posedness issue for the cubic nonlinear Schr\"odinger equation. In particular, we prove norm inflation based at every initial condition in negative Sobolev spaces below or at the scaling critical regularity.

A Remark on Norm Inflation with General Initial Data for the Cubic Nonlinear Schr&ouml;dinger Equations in Negative Sobolev Spaces

A Remark on Norm Inflation with General Initial Data for the Cubic Nonlinear Schrödinger Equations in Negative Sobolev Spaces