Virial identity

A virial identity describes the time derivatives of a spatial moment of a solution to a dispersive equation. For the mass-critical nonlinear Schrödinger equation, the second derivative of the variance is a constant multiple of the conserved energy.

A localized virial identity differentiates a weighted mass $\int \chi |u{|}^2$ and its associated momentum. Compactly supported or flattened weights retain the coercive interior contribution while replacing an infinite-variance assumption by controllable 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.