= Hartree mass and energy conservation
{c}
= Hartree mass conservation
{c}
{synonym}
= Hartree energy conservation
{c}
{synonym}
For sufficiently regular, decaying solutions of the <Gravitational Hartree equation> with finite displayed integrals, <integration by parts> gives the conserved quantities
$$
M(u)=\int|u|^2,
\qquad
E(u)=\frac12\int|\nabla u|^2-\frac14\int|\nabla\phi|^2.
$$
The self-adjointness of the Newtonian convolution turns the time derivative of the potential energy into $\operatorname{Re}\int\phi u\overline{u_t}$, which cancels the derivative of the kinetic energy by the equation.
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