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On singularity formation for the \(L^{2}\)-critical boson star equation. (English) Zbl 1235.35231

Summary: We prove a general, non-perturbative result about finite-time blowup solutions for the \(L^2\)-critical Boson star equation \[ i\partial_t u=\sqrt{-\Delta+m^2}u-(|x|^{-1}*|u|^2)u, \] 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 \(0<T<+\infty\), we show that \(u(t,\cdot)\) has a unique weak limit in \(L^2\) and that \(|u(t,\cdot)|^2\) has a unique weak limit in the sense of measures as \(t\to T^-\). 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 \(L^2(\{|x|\geq R\})\) as \(t\to T^-\) 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.

MSC:

35Q40 PDEs in connection with quantum mechanics
35B44 Blow-up in context of PDEs
35Q75 PDEs in connection with relativity and gravitational theory
81V17 Gravitational interaction in quantum theory
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