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The question of whether the hyper-dissipative (HD) Napier-Stokes (NS) systemcan exhibit spontaneous formation of singularities in the super-criticalregime--the hyper-diffusion being generated by a fractional power of theLaplacian, say $\beta$, confined to interval $\bigl(1, \frac{5}{4}\bigr)$--hasbeen a major open problem in the mathematical fluid dynamics since thefoundational work of J.L. Lions in 1960s. In this work, an evidence ofcriticality of the Laplacian is presented, more precisely, a class of plausibleblow-up scenarios is ruled out as soon as $\beta$ is greater than one. Whilethe framework is based on the scale of sparseness of the super-level sets ofthe positive and negative parts of the components of the higher-orderderivatives of the velocity recently introduced by the authors, a major noveltyin the current work is classification of the HD flows near a potentialspatiotemporal singularity in two main categories, homogeneous (the caseconsistent with a near-steady behavior) and non-homogenous (the case consistentwith the formation and decay of turbulence). The main theorem states that inthe non-homogeneous case any $\beta$ greater than one prevents a singularity.In order to illustrate the impact of this result in a methodology-free setting,a two-parameter family of dynamically rescaled blow-up profiles is considered,and it is shown that as soon as $\beta$ is greater than one, a new region inthe parameter space is ruled out. More importantly, the region is aneighborhood (in the parameter space) of the self-similar profile, i.e., theapproximately self-similar blow-up, a prime suspect in possible singularityformation, is ruled out for all HD NS models.

Time-Global Regularity of the Navier-Stokes System with Hyper-Dissipation--Turbulent Scenario