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We establish quantitative blow-up criteria below the scaling threshold forradially symmetric solutions to the defocusing nonlinear Schr\"odinger equationwith nonlinearity $|u|^6u$. This provides to our knowledge the first genericresults distinguishing potential blow-up solutions of the defocusing equationfrom many of the known examples of blow-up in the focusing case. Our main toolis a quantitative version of a result showing that uniform bounds on$L^2$-based critical Sobolev norms imply scattering estimates.  As another application of our techniques, we establish a variant which allowsfor slow growth in the critical norm. We show that if the critical Sobolev normon compact time intervals is controlled by a slowly growing quantity dependingon the Stricharz norm, then the solution can be extended globally in time, witha corresponding scattering estimate.

Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds