 Hartree mass and energy conservation

ID: hartree-mass-and-energy-conservation

Codex  0 Created 2026-09-28 Updated 2026-11-05

For sufficiently regular, decaying solutions of the Gravitational Hartree equation with finite displayed integrals, integration by parts gives the conserved quantities

M (u) = \int ∣ u ∣^{2}, E (u) = \frac{1}{2} \int ∣\nabla u ∣^{2} - \frac{1}{4} \int ∣\nabla ϕ ∣^{2} .

The self-adjointness of the Newtonian convolution turns the time derivative of the potential energy into

Re \int ϕ u \overline{u_{t}}

, which cancels the derivative of the kinetic energy by the equation.

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