On Singularity formation for the L^2-critical Boson star equation

TL;DR

This paper proves new results on finite-time blowup solutions for the L^2-critical boson star equation, showing measure convergence, minimal mass concentration, and radial solution behavior, advancing understanding of gravitational collapse in quantum models.

Contribution

It establishes measure convergence, minimal mass concentration, and radial convergence properties for blowup solutions, providing a significant step towards the large data blowup conjecture for this equation.

Findings

Weak limits in L^2 and measures are unique for blowup solutions.

Radial solutions converge strongly away from the origin.

Results extend to other L^2-critical gravitational collapse models.

Abstract

We prove a general, non-perturbative result about finite-time blowup solutions for the $L^2$-critical boson star equation $i\partial_t u = \sqrt{-\Delta+m^2} \, u - (|x|^{-1} \ast |u|^2) u$ in 3 space dimensions. Under the sole assumption that the solution blows up in $H^{1/2}$ at finite time, 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. 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 behavior of $u$. As the second main result, we prove that any radial finite-time blowup solution $u$ converges strongly in $L^2$ away from the origin. For radial solutions, this result establishes a large data blowup conjecture for the $L^2$-critical boson star equation, similar to a conjecture which was originally formulated by F. Merle and P. Raphael for the $L^2$-critical nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704]. We also discuss some extensions of our results to other $L^2$-critical theories of gravitational collapse, in particular to critical Hartree-type equations.