On singularity formation for the L²-critical Boson star equation

We prove a general, non-perturbative result about finite-time blowup solutions for the L²-critical Boson star equation in d = 3 space dimensions. Under the sole assumption that u = u(t, x) blows up in the energy space H^1/2 at finite time 0 < T < +∞, we show that u(t, ) has a unique weak limit in L² and that |u(t, )|² has a unique weak limit in the sense of measures as t → T^-. Moreover, we prove that the limiting measure exhibits minimal mass concentration. A central ingredient used in the proof is a 'finite speed of propagation' property, which puts a strong rigidity on the blowup behaviour of u = u(t, x). As the second main result, we prove that any radial finite-time blowup solution u = u(t, |x|) converges strongly in L²({|x| ≥ R}) as t → T^- for any R > 0. For radial solutions, this result establishes a large data blowup conjecture for the L²-critical Boson star equation. We also discuss some extensions of our results to other L²-critical theories of gravitational collapse, in particular to critical Hartree-type equations.