Asymptotic Criticality of the Navier-Stokes Regularity Problem

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 supe...

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Bibliographic Details
Published inarXiv.org
Main Authors Grujic, Zoran, Xu, Liaosha
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 21.12.2023
Subjects
Computational fluid dynamics
Fluid flow
Mathematical analysis
Navier-Stokes equations
Regularity
Singularities
Velocity distribution
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Summary: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 \emph{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 \emph{a priori} bound vanishes as the order of the derivative goes to infinity.
ISSN:2331-8422