 Localized virial identity

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.