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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.

Let $0<\lambda <d$ and $1<p,q<\infty$ satisfy $1/p+1/q+\lambda /d=2$. The Hardy–Littlewood–Sobolev inequality states thatHere the right side uses Lp norms, and the constant is independent of $f,g$. The exponents are linked by the displayed dimensional relation; they are not a conjugate exponents pair. This is a bound for a singular convolution rather than a pointwise bound on the kernel.^[ref]

For $d=3$, $\lambda =1$ and $p=q=6/5$, applying it to $f=g=|u{|}^2$ givesThis is the potential-energy estimate used for the Gravitational Hartree equation. A Gagliardo-Nirenberg interpolation inequality then relates that intermediate Lp norm to mass and the gradient L2 norm.

The Hartree equations are self-consistent one-particle equations obtained by approximating a many-body wave function by a tensor product of orbital wave functions. Each orbital sees the potential averaged over the other particles' probability densities; attractive Newtonian mean-field variants include the Gravitational Hartree equation.

For sufficiently regular, decaying solutions of the Gravitational Hartree equation with finite displayed integrals, integration by parts gives the conserved quantitiesThe self-adjointness of the Newtonian convolution turns the time derivative of the potential energy into $\mathrm{Re}\int \varphi u\overline{u_t}$, which cancels the derivative of the kinetic energy by the equation.

The Gravitational Hartree equation can be writtenwhere $\varphi$ is real. Consequentlybecause both integrals multiplied by $i$ are purely imaginary after integration by parts. Hence Hartree mass conservation holds.