Existence or exclusion of self-similar blow-up in three-dimensional energy-critical NLS

Determine whether self-similar blow-up exists for the three-dimensional energy-critical nonlinear Schrödinger equation, or prove that such blow-up is impossible.

Background

The paper discusses several known finite-time blow-up regimes for the three-dimensional energy-critical nonlinear Schrödinger equation, including standing-ring log-log blow-up and single-bubble type II blow-up. Despite these constructions, the status of self-similar blow-up—whether it occurs or is excluded—remains unresolved.

References

For the energy-critical NLS eq:NLS in $d=3$, log-log blow-up has been constructed in the standing-ring regime , while the existence or exclusion of self-similar blow-up remains open.

— Smooth radial quantized Type II blow-up for the three-dimensional energy-critical nonlinear Schrödinger equation  (2610.26127 - Kim, 19 Oct 2026) in Section 1, subsection “Discussion of the results,” Remark “Comparison with mass-critical NLS”