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 Incoming links: Low-Regularity Global Solutions to Nonlinear Dispersive Equations by Terence Tao

Energy conservation for the nonlinear Schrödinger equation Created 2026-09-24 Updated 2026-11-05

For the focusing nonlinear Schrödinger equation on

R^{d}

i u_{t} + Δ u + ∣ u ∣^{p - 1} u = 0, p > 1,

(1)

the energy functional is

E (u) = \frac{1}{2} \int_{R^{d}} ∣\nabla u ∣^{2} d x - \frac{1}{p + 1} \int_{R^{d}} ∣ u ∣^{p + 1} d x .

(2)

For a sufficiently smooth solution with decay permitting differentiation under the integral sign and integration by parts, it is constant in time. Indeed,

\frac{d E}{d t} = Re \int_{R^{d}} (- Δ u - ∣ u ∣^{p - 1} u) \overline{u_{t}} d x = Re \int_{R^{d}} i ∣ u_{t} ∣^{2} d x = 0.

(3)

The cancellation is the Hamiltonian conservation mechanism: the evolution rotates the energy gradient by multiplication by

i

. The focusing sign makes the nonlinear contribution negative, so conserved energy alone need not bound the L2 norm of the gradient.^[ref]

For less regular solutions the displayed computation is only formal. To inherit the identity, approximate by smooth solutions and justify convergence of both energy terms. Convergence in Sobolev space

H^{1}

controls the nonlinear term through a suitable Sobolev embedding only in the applicable exponent range, or with additional integrability. An arbitrary weak solution must not be assumed to satisfy the identity solely because the smooth equation does.

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