Diffusion of Lagrangian invariants in the Navier-Stokes equations

  • Peter Constantin
Conference paper
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 71)


The incompressible Euler equations can be written as the active vector system
$$ (\partial _t + u \cdot \nabla )A = 0$$
where u=W[A] is given by the Weber formula
$$ W[A] = P\{ (\nabla A)^* \upsilon \}$$
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.


Analytic Norm Particle Path Active Vector Vortex Method Cauchy Formula 
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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Peter Constantin
    • 1
  1. 1.Department of Mathematics ChicagoThe University of ChicagoUSA

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