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

The gravitational Hartree equation couples a wave function to its attractive Newtonian potential:

The gravitational Hartree flow conservesThe 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.