2022/03/15 by Nicolás Lerner, Lerner, Nicolas, François Vigneron +1
Chemical Engineering · Engineering · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Rheology and Fluid Dynamics Studies
paper · pdf · doi:10.48550/arxiv.2203.07950
openalex publication_date 2022/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate some geometric properties of the curl operator, based on its diagonalizationand its expression as a non-local symmetry of the pseudo-derivative (-Δ)1/2 among divergence-free vector fieldswith finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectorsof (-Δ)-1/2curl.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-productand 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 caseof a finite-time blow-up, both spin-definite components of the flow will explose 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 ∫0T∫ℝ3det(curl u, u, (-Δ)θ u) and their spin-definite counterparts, which drive the enstrophy and, more generally, are responsible forthe regularity of the flow and the emergence of singularities or quasi-singularities.As such, they are at the core of turbulence phenomena.