On singularity formation for theL2-critical Boson star equation

We prove a general, non-perturbative result about finite-time blowup solutions for the L 2 -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 L 2 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 L 2 ({| x | ⩾ R }) as t → T − for any R > 0. For radial solutions, this result establishes a large data blowup conjecture for the L 2 -critical Boson star equation. We also discuss some extensions of our results to other L 2 -critical theories of gravitational collapse, in particular to critical Hartree-type equations.