Gravitational Hartree equation ($i{u}_t+\Delta u-\varphi u=0$)

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.

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.

In three dimensions, the Hardy–Littlewood–Sobolev inequality and Gagliardo-Nirenberg interpolation inequality giveConserved mass and energy therefore bound the $H^1$ norm by Young inequality, and the blowup alternative gives global existence.

For a finite-variance solution in four dimensions,satisfies $I^{\prime \prime }(t)=16E(u)$. The identity uses the degree $-2$ homogeneity of the Newtonian kernel. Negative-energy finite-variance data consequently blow up in finite time by the convexity argument.