= Blowup alternative for the nonlinear Schrödinger equation
{c}
For the <mass-critical focusing nonlinear Schrödinger equation>, let $u\in C([0,T_+);H^1(\mathbb R^d))$ be the maximal forward solution in the standard local $H^1$ theory. Either $T_+=\infty$, or
$$
\lim_{t\uparrow T_+}\|u(t)\|_{H^1}=\infty.
$$
Because <Mass conservation for the nonlinear Schrödinger equation> fixes $\|u(t)\|_2$, this is equivalent to divergence of $\|\nabla u(t)\|_2$.
The proof uses a local existence time bounded below for data in a bounded <H1 norm> ball. If a sequence $t_n\uparrow T_+$ had uniformly bounded <H1 norms>, restart the equation at $t_n$ with that common time interval. Uniqueness joins the restarted solution to the old one and extends it beyond $T_+$, contradicting maximality.
The dependence of lifespan on the norm is a hypothesis of this argument. Other critical solution theories may use a spacetime control norm instead, so the stated <H1 norm> conclusion is attached here to the mass-critical equation and its $H^1$-subcritical local theory.<Nonlinear dispersive equations: local and global analysis by Terence Tao>[Section 3.3, local theory and controlling norms; section 3.4, continuation and conservation]{ref}
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