An Algebraic Reduction of the ‘Scaling Gap’ in the Navier–Stokes Regularity Problem
It is shown—within a mathematical framework based on the suitably defined scale of sparseness of the super-level sets of the positive and negative parts of the vorticity components, and in the context of a blow-up-type argument—that the ever-resisting ‘scaling gap’, that is, the scaling distance between a regularity criterion and a corresponding a priori bound (shortly, a measure of the super-criticality of the three dimensional Navier–Stokes regularity problem), can be reduced by an algebraic factor; since (independent) fundamental works of Ladyzhenskaya, Prodi and Serrin as well as Kato and Fujita in 1960s, all the reductions have been logarithmic in nature, regardless of the functional set up utilized. More precisely, it is shown that it is possible to obtain an a priori bound that is algebraically better than the energy-level bound, while keeping the corresponding regularity criterion at the same level as all the classical regularity criteria. The mathematics presented was inspired by morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of turbulent flows, as well as by the physics of turbulent cascades and turbulent dissipation.
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The work of A.F. is supported in part by the National Science Foundation Grant DMS-1418911. The work of Z.G. is supported in part by the National Science Foundation Grant DMS-1515805 and the Lundbeck Foundation Grant R217-2016-446 (a collaborative grant with the NBI and Aarhus). Z.G. thanks the Program in Applied and Computational Mathematics at Princeton for their hospitality in Spring 2017 when the paper was finalized. The authors thank the referee for the constructive comments.
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