Asymptotic Criticality of the Navier-Stokes Regularity Problem
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866912240329818112 |
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| author | Grujic, Zoran Xu, Liaosha |
| author_facet | Grujic, Zoran Xu, Liaosha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1911_00974 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Asymptotic Criticality of the Navier-Stokes Regularity Problem Grujic, Zoran Xu, Liaosha Analysis of PDEs Mathematical Physics 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. |
| title | Asymptotic Criticality of the Navier-Stokes Regularity Problem |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/1911.00974 |