The <Gravitational Hartree equation> can be written
$$
u_t=i\Delta u-i\phi u,
$$
where $\phi$ is real. Consequently
$$
\frac d{dt}\int_{\mathbb R^N}|u|^2\,dx
=2\operatorname{Re}\int_{\mathbb R^N}u_t\overline u\,dx
=2\operatorname{Re}\left(
i\int\Delta u\,\overline u-i\int\phi|u|^2
\right)=0,
$$
because both integrals multiplied by $i$ are purely imaginary after <integration by parts>. Hence <Hartree mass conservation> holds.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 2 / 1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-154/2/1/solution