Abstract:We prove norm inflation for the cubic hyperbolic nonlinear Schrödinger equation in $H^s(\mathbb T^2)$ for every $s\in(-\infty,0)\cup(0,\frac12]$. The scaling-critical point $s=0$ is excluded by conservation of the $L^2$ norm. The strong ill-posedness below and above the scaling-critical point arises from two completely different mechanisms. Particularly in the scaling-subcritical regime, this dynamical instability stems from the hyperbolic nature. Together with the local well-posedness result in \cite{WangHNLS}, this gives a sharp dichotomy away from the mass space $L^2(\mathbb T^2)$: local well-posedness holds for $s>\frac12$, whereas norm inflation occurs for all $s\le \frac12$ with $s\ne0$.
From: Yuzhao Wang [view email]
[v1]
Wed, 17 Jun 2026 17:31:21 UTC (21 KB)
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