= Negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation
For a finite-variance solution of the <mass-critical focusing nonlinear Schrödinger equation>, the <virial identity> gives
$$
\frac{d^2}{dt^2}\int_{\mathbb R^d}|x|^2|u(t,x)|^2\,dx=16E(u_0).
$$
If $E(u_0)<0$, the right-hand side is a negative constant. The nonnegative variance would then become negative in finite time if the solution remained regular, so the solution blows up in finite time.
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