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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)

Abstract

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.

Keywords

Analytic Norm Particle Path Active Vector Vortex Method Cauchy Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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