摘要
We investigate some geometric properties of the curl operator,based on its diagonalization and its expression as a non-local symmetry of the pseudoderivative(-△)^(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 crossproduct 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 ∫_(0)^(T) ∫_(R^(3))det(curlu,u,(-△)^(θ)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.
