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{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.

Global H1 solutions of the three-dimensional gravitational Hartree equation

In three dimensions, the <Hardy–Littlewood–Sobolev inequality> applied to the Newtonian kernel gives
$$
\int_{\mathbb R^3}|\nabla\phi|^2
=-\int\phi|u|^2
\lesssim\||u|^2\|_{6/5}^2
=\|u\|_{12/5}^4.
$$
The <Gagliardo-Nirenberg interpolation inequality> then gives
$$
\|u\|_{12/5}^4
\lesssim\|u\|_2^3\|\nabla u\|_2.
$$
Writing $X=\|\nabla u\|_2$ and using the conserved mass, conservation of energy implies
$$
E(u_0)\geq\frac12X^2-C(u_0)X.
$$
Thus $X$ remains bounded. Together with the conserved $L^2$ norm this bounds $\|u(t)\|_{H^1}$, and the supplied blowup criterion proves <Global H1 solutions of the three-dimensional gravitational Hartree equation>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 2 / 3 / Solution

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For $0<\lambda<d$ and suitable conjugate exponents, the Hardy–Littlewood–Sobolev inequality bounds convolution with the Riesz kernel $|x|^{-\lambda}$. One useful form is
$$
\iint_{\mathbb R^d\times\mathbb R^d}
\frac{f(x)g(y)}{|x-y|^\lambda}\,dx\,dy
\lesssim\|f\|_p\|g\|_q,
\qquad
\frac1p+\frac1q+\frac\lambda d=2.
$$

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 Incoming links: Hardy–Littlewood–Sobolev inequality