### Abstract

The incompressible Euler equations can be written as the active vector system (∂_{t}gt; + u. ∇) A = 0 where u = W[A] is given by the Weber formula W[A] = P{(∇A)^{*}v} 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_{o} 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.

Original language | English (US) |
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Title of host publication | Tubes, Sheets and Singularities in Fluid Dynamics |

Publisher | Kluwer Academic Publishers |

Pages | 285-294 |

Number of pages | 10 |

ISBN (Print) | 1402009801, 9781402009808 |

DOIs | |

State | Published - 2004 |

### Publication series

Name | Fluid Mechanics and its Applications |
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Volume | 71 |

ISSN (Print) | 0926-5112 |

### All Science Journal Classification (ASJC) codes

- Mechanics of Materials
- Mechanical Engineering
- Fluid Flow and Transfer Processes

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## Cite this

Constantin, P. (2004). Diffusion of Lagrangian invariants in the Navier-Stokes equations. In

*Tubes, Sheets and Singularities in Fluid Dynamics*(pp. 285-294). (Fluid Mechanics and its Applications; Vol. 71). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-48420-x_35