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 Séminaires
We investigate some geometric properties of the curl operator, based on its diagonalization and its expression as a nonlocal symmetry of the pseudoderivative (−∆)^{1/2} among divergencefree vector fields with finite energy. In this context, we introduce the notion of spindefinite fields, i.e. eigenvectors of (−∆)^{−1/2} curl. The two spindefinite components of a general 3D incompressible flow untangle the righthanded motion from the lefthanded one.
The nonlinearity of NavierStokes has the structure of a crossproduct. In the case of a finitetime blowup, both spindefinite 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 nonlocal determinants det(curl(u),u,(−∆)^\theta u), which drive enstrophy and are responsible for the regularity of the flow or the emergence of singularities or quasisingularities. As such, they are at the core of turbulence phenomena.