{"title":"Navier-Stokes系统旋度算子的一些性质及其结果","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":"{\"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}","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}
On Some Properties of the Curl Operator and Their Consequences for the Navier-Stokes System
. 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.