Global H1 solutions of the three-dimensional gravitational Hartree equation
= Global H1 solutions of the three-dimensional gravitational Hartree equation
{c}
{title2=$N=3$}
In three dimensions, the <Hardy–Littlewood–Sobolev inequality> and <Gagliardo-Nirenberg interpolation inequality> give
$$
\int|\nabla\phi|^2
\lesssim\|u\|_{12/5}^4
\lesssim\|u\|_2^3\|\nabla u\|_2.
$$
Conserved mass and energy therefore bound the $H^1$ norm by <Young inequality>, and the blowup alternative gives global existence.