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Abstract The question of whether the hyper-dissipative (HD) Navier-Stokes (NS) system can exhibit spontaneous formation of singularities in the super-critical regime–the hyperviscous effects being represented by a fractional power of the Laplacian, say$$\beta $$ $\beta$, confined to interval$$\bigl (1, \frac{5}{4}\bigr )$$ $(1,\frac{5}{4})$–has been a major open problem in the mathematical fluid dynamics since the foundational work of J.L. Lions in 1960s. In this work, an evidence of criticality of the Laplacian is presented, more precisely, a class of plausible blow-up scenarios is ruled out as soon as$$\beta $$ $\beta$is greater than one. While the framework is based on the ‘scale of sparseness’ of the super-level sets of the positive and negative parts of the components of the higher-order derivatives of the velocity previously introduced by the authors, a major novelty in the current work is classification of the HD flows near a potential spatiotemporal singularity in two main categories, ‘homogeneous’ (the case consistent with a near-steady behavior) and ‘non-homogenous’ (the case consistent with the formation and decay of turbulence). The main theorem states that in the non-homogeneous case any$$\beta $$ $\beta$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$$\beta $$ $\beta$is greater than one, a new region in the parameter space is ruled out. More importantly, the region is a neighborhood (in the parameter space) of the self-similar profile, i.e., the approximately self-similar blow-up, a prime suspect in possible singularity formation, is ruled out for all HD NS models. 

Abstract The problem of global-in-time regularity for the 3D Navier-Stokes equations, i.e., the question of whether a smooth flow can exhibit spontaneous formation of singularities, is a fundamental open problem in mathematical physics. Due to the super-criticality of the equations, the problem has been super-critical in the sense that there has been a ‘scaling gap’ between any regularity criterion and the corresponding a priori bound (regardless of the functional setup utilized). The purpose of this work is to present a mathematical framework-based on a suitably defined ‘scale of sparseness’ of the super-level sets of the positive and negative parts of the components of the higher-order spatial derivatives of the velocity field—in which the scaling gap between the regularity class and the corresponding a priori bound vanishes as the order of the derivative goes to infinity.