Let $\phi \in H^s(\mathbb{T})$. The Cauchy problem for

\begin{equation} iu_t + \Delta u = |u|^2 u \end{equation}

with $u(0) = \phi$ is well-posed for $s \geq 0$.

Let $\phi \in H^s(\mathbb{T})$. The Cauchy problem for

\begin{equation} iu_t + \Delta u = |u|^2 u \end{equation}

with $u(0) = \phi$ is well-posed for $s \geq 0$.