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

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 Schrödinger 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.

24 pages. Accepted in Nonlinearity

References in corpus (3)

• Minimizers for the Hartree-Fock-Bogoliubov Theory of Neutron Stars and White Dwarfs
• On Blowup for time-dependent generalized Hartree-Fock equations
• Stellar Collapse in the Time Dependent Hartree-Fock Approximation

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• Stabilization against collapse of 2D attractive Bose-Einstein condensates with repulsive, three-body interactions
• Many-Body Blow-Up Profile of Boson Stars with External Potentials
• Validity of Bogoliubov's approximation for translation-invariant Bose gases
• Multiplicity and asymptotics of standing waves for the energy critical half-wave
• The relativistic mean-field equations of the atomic nucleus
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• Existence and non-existence of normalized solutions for a nonlinear fractional Schrödinger system
• Blow-up profile of neutron stars in the Hartree-Fock-Bogoliubov theory
• Boson Stars with Long-range Perturbations
• The fractional in time Schrödinger equation with a Hartree perturbation
• On the stability of 2D dipolar Bose-Einstein condensates