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The Gravitational Hartree equation couples a complex wave function to the attractive Newtonian gravitational potential generated by its own squared modulus. In dimension $N\ge 3$, a dimensionless convention iswhere $C_N=(N-2)|S^{N-1}|$ and $|S^{N-1}|$ is the area of the unit sphere. The minus sign in the fundamental solution makes $\varphi \le 0$ for nonnegative density, with the potential selected to decay at infinity when the convolution permits it.

The convolution makes the nonlinearity nonlocal: changing the density at one point changes the potential elsewhere. The Newtonian kernel and its dimension differ from a local cubic nonlinear Schrödinger equation. In three dimensions, the Hardy–Littlewood–Sobolev inequality controls the interaction energy and leads to Global H1 solutions of the three-dimensional gravitational Hartree equation.

In three dimensions, the Hardy–Littlewood–Sobolev inequality applied to the Newtonian kernel givesThe Gagliardo-Nirenberg interpolation inequality then givesWriting $X=\Vert \nabla u{\Vert }_2$ and using the conserved mass, conservation of energy impliesThus $X$ remains bounded. Together with the conserved $L^2$ norm this bounds $\Vert u(t){\Vert }_{H^1}$, and the supplied blowup criterion proves Global H1 solutions of the three-dimensional gravitational Hartree equation.