Hartree mass and energy conservation

For sufficiently regular, decaying solutions of the Gravitational Hartree equation with finite displayed integrals, integration by parts gives the conserved quantitiesThe self-adjointness of the Newtonian convolution turns the time derivative of the potential energy into $\mathrm{Re}\int \varphi u\overline{u_t}$, which cancels the derivative of the kinetic energy by the equation.