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Volume 38, Issue 4
On Some Properties of the Curl Operator and Their Consequences for the Navier-Stokes System

Nicolas Lerner & François Vigneron

Commun. Math. Res., 38 (2022), pp. 449-497.

Published online: 2022-10

[An open-access article; the PDF is free to any online user.]

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  • 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 $(−\Delta)^{1/2}$ among divergence-free vector fields with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of $(−\Delta)^{−1/2}$ 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  $$\int_0^T\int_{\mathbb{R}^3}{\rm det}({\rm curl}u,u,(-\Delta)^{\theta}u)$$ and their spin-definite counterparts, which drive the enstrophy and, more generally, 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.

  • AMS Subject Headings

35Q30, 35B06, 76D05, 76F02

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-38-449, author = {Lerner , Nicolas and Vigneron , François}, title = {On Some Properties of the Curl Operator and Their Consequences for the Navier-Stokes System}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {38}, number = {4}, pages = {449--497}, 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 $(−\Delta)^{1/2}$ among divergence-free vector fields with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of $(−\Delta)^{−1/2}$ 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  $$\int_0^T\int_{\mathbb{R}^3}{\rm det}({\rm curl}u,u,(-\Delta)^{\theta}u)$$ and their spin-definite counterparts, which drive the enstrophy and, more generally, 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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0106}, url = {http://global-sci.org/intro/article_detail/cmr/21067.html} }
TY - JOUR T1 - On Some Properties of the Curl Operator and Their Consequences for the Navier-Stokes System AU - Lerner , Nicolas AU - Vigneron , François JO - Communications in Mathematical Research VL - 4 SP - 449 EP - 497 PY - 2022 DA - 2022/10 SN - 38 DO - http://doi.org/10.4208/cmr.2021-0106 UR - https://global-sci.org/intro/article_detail/cmr/21067.html KW - Navier-Stokes, vorticity, hydrodynamic spin, critical determinants, turbulence. AB -

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 $(−\Delta)^{1/2}$ among divergence-free vector fields with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of $(−\Delta)^{−1/2}$ 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  $$\int_0^T\int_{\mathbb{R}^3}{\rm det}({\rm curl}u,u,(-\Delta)^{\theta}u)$$ and their spin-definite counterparts, which drive the enstrophy and, more generally, 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.

Nicolas Lerner & François Vigneron. (2022). On Some Properties of the Curl Operator and Their Consequences for the Navier-Stokes System. Communications in Mathematical Research . 38 (4). 449-497. doi:10.4208/cmr.2021-0106
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