Abstract
The incompressible Euler equations can be written as the active vector system
where u=W[A] is given by the Weber formula
in terms of the gradient of A and the passive field v=u 0 (A). (P is the projector on the divergence-free part.) The initial data is A(x,0)=x, so for short times this is a distortion of the identity map. After a short time one obtains a new u and starts again from the identity map, using the new u instead of u 0 in the Weber formula. The viscous Navier-Stokes equations admit the same representation, with a diffusive back-to-labels map A and a v that is no longer passive.
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© 2002 Kluwer Academic Publishers
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Constantin, P. (2002). Diffusion of Lagrangian invariants in the Navier-Stokes equations. In: Bajer, K., Moffatt, H.K. (eds) Tubes, Sheets and Singularities in Fluid Dynamics. Fluid Mechanics and Its Applications, vol 71. Springer, Dordrecht. https://doi.org/10.1007/0-306-48420-X_35
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DOI: https://doi.org/10.1007/0-306-48420-X_35
Publisher Name: Springer, Dordrecht
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