Virial identity

For a sufficiently regular solution of the mass-critical focusing nonlinear Schrödinger equation with finite variance, the variance and its first derivative areA second differentiation and integration by parts give the virial identityThese constants use Laplacian coefficient one in the evolution equation. Energy conservation makes the second derivative constant. If the energy is negative, the resulting quadratic expression for $I(t)$ eventually becomes negative, contradicting its definition as a nonnegative integral; this proves negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation under the finite-variance hypotheses.

The weighted moment condition is additional to an H1 norm bound. A localized virial identity replaces the unbounded weight by a smooth cutoff when only tail control is available.

For $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, a smooth real weight $\chi$ defines the localized mass $I_{\chi }(t)=\int \chi |u{|}^2$. For a sufficiently regular decaying solution and a weight with the required integrability,andThese formulas follow by substituting the nonlinear Schrödinger equation and integration by parts. The Hessian controls the gradient contribution; the fourth derivative term records variation of the cutoff.

Choose ${\chi }_R(x)=R^2\chi (x/R)$, with $\chi (x)=|x{|}^2$ near zero and constant outside a larger ball. The interior terms reproduce the virial identity, while the remaining terms are supported in the transition region or controlled by spatial tails. A bounded weight makes $I_{{\chi }_R}$ meaningful for finite mass without requiring finite variance. A blowup proof still needs estimates that control those tail errors.

For a finite-variance solution of the mass-critical focusing nonlinear Schrödinger equation, the virial identity givesIf $E(u_0)<0$, 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.