Rigidity classification of single-bubble blow-up rates

Characterize all possible single-bubble blow-up rates for finite-time solutions of the three-dimensional energy-critical nonlinear Schrödinger equation, thereby establishing a rigidity classification.

Background

The paper constructs smooth radial single-bubble type II blow-up solutions with quantized rates and contrasts them with previously constructed single-bubble solutions having a continuum of blow-up rates. The coexistence of these rate families leaves unresolved the question of which rates are possible in general. A rigidity theorem classifying all single-bubble blow-up rates is therefore explicitly identified as an open problem.

References

Nevertheless, a rigidity result classifying the possible single-bubble blow-up rates 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”