Norm inflation for a higher-order nonlinear Schr\"odinger equation with a derivative on the circle

We consider a periodic higher-order nonlinear Schr\"odinger equation with the nonlinearity uᵏ ∂ₓ u, where k is a natural number. We prove the norm inflation in a subspace of the Sobolev space Hˢ(T) for any s ∈ R. In particular, the Cauchy problem is ill-posed in Hˢ(T) for any s ∈ R.