= Nonlinear Schrödinger blowup analysis
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<Nonlinear Schrödinger blowup analysis> asks whether a solution remains in its local well-posedness class or concentrates so strongly that continuation fails in finite time. For the <mass-critical focusing nonlinear Schrödinger equation> in the standard $H^1$ theory, finite-time failure forces the gradient <L2 norm> to diverge, even though <Mass conservation for the nonlinear Schrödinger equation> keeps the total squared <L2 norm> constant.
Concentration and total size are different. The mass-preserving scaling $u_\lambda(x)=\lambda^{d/2}u(\lambda x)$ leaves $\|u_\lambda\|_2=\|u\|_2$ while $\|\nabla u_\lambda\|_2=\lambda\|\nabla u\|_2$. Thus a fixed amount of mass can occupy smaller spatial scales with larger gradients.
The <Sharp Gagliardo-Nirenberg inequality> gives <coercivity below the NLS ground-state mass>, while a <virial identity> can force finite-time failure for suitable negative-energy, finite-variance data. Near the threshold, <profile decomposition modulo translations> separates a concentrating profile from dispersive remainder. These arguments track the structure of a solution rather than merely the maximum of a plotted wave.
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