{"title":"On Some Properties of the Curl Operator and Their Consequences for the Navier-Stokes System","authors":"N. Lerner, Franccois Vigneron","doi":"10.4208/cmr.2021-0106","DOIUrl":null,"url":null,"abstract":". We investigate some geometric properties of the curl operator, based on its diagonalization and its expression as a non-local symmetry of the pseudo-derivative ( − ∆ ) 1/2 among divergence-free vector fields with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of ( − ∆ ) − 1/2 curl. The two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed one. Having observed that the non-linearity of Navier-Stokes has the structure of a cross-product and its weak (distributional) form is a determinant that involves the vorticity, the velocity and a test function, we revisit the conservation of energy and the balance of helicity in a geometrical fashion. We show that in the case of a finite-time blow-up, both spin-definite components of the flow will explode simultaneously and with equal rates, i.e. singularities in 3D are the result of a conflict of spin, which is impossible in the poorer geometry of 2D flows. We investigate the role of the local and non-local determinants and their spin-definite counterparts, which drive the enstrophy and, more gen-erally, are responsible for the regularity of the flow and the emergence of singularities or quasi-singularities. As such, they are at the core of turbulence phenomena.","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"数学研究通讯","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4208/cmr.2021-0106","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
. We investigate some geometric properties of the curl operator, based on its diagonalization and its expression as a non-local symmetry of the pseudo-derivative ( − ∆ ) 1/2 among divergence-free vector fields with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of ( − ∆ ) − 1/2 curl. The two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed one. Having observed that the non-linearity of Navier-Stokes has the structure of a cross-product and its weak (distributional) form is a determinant that involves the vorticity, the velocity and a test function, we revisit the conservation of energy and the balance of helicity in a geometrical fashion. We show that in the case of a finite-time blow-up, both spin-definite components of the flow will explode simultaneously and with equal rates, i.e. singularities in 3D are the result of a conflict of spin, which is impossible in the poorer geometry of 2D flows. We investigate the role of the local and non-local determinants and their spin-definite counterparts, which drive the enstrophy and, more gen-erally, are responsible for the regularity of the flow and the emergence of singularities or quasi-singularities. As such, they are at the core of turbulence phenomena.