Skip to main content
SpringerLink
Log in

Diffusion of Lagrangian invariants in the Navier-Stokes equations

  • Conference paper
Book cover Tubes, Sheets and Singularities in Fluid Dynamics

Part of the book series: Fluid Mechanics and Its Applications ((FMIA,volume 71))

  • 596 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Arnol’d, V. I. 1984 Mathematical methods of classical mechanics, Springer-Verlag, GTM 60 New York.

    Google Scholar 

  • Beale, J. T., Kato, T., Majda, A. 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys.94, 61–66.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Chorin, A. 1973 Numerical study of slightly viscous flow, J. Fluid. Mech.57, 785–796.

    Article  MathSciNet  ADS  Google Scholar 

  • Constantn, P. 1994, Geometric statistics in turbulence, Siam Review, 36, 73–98.

    Article  MathSciNet  Google Scholar 

  • Constantin, P. 2001 (a) An Eulerian-Lagrangian approach for incompressible fluids: Local theory, J. Amer. Math. Soc.14, 263–278.

    Article  MathSciNet  MATH  Google Scholar 

  • Constantin, P. 2001(b), An Eulerian-Lagrangian approach for the Navier-Stokes equations, Commun. Math. Phys.216, 663–686.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Constantin, P. 2002, Near Identity Transformations for the Navier-Stokes Equations, preprint xxx.lanl.gov/list/math.AP/0112128.

    Google Scholar 

  • John, P. 1991 Partial Differntial Equations Applied Math. Sciences 1, Springer-Verlag, New York.

    Google Scholar 

  • Moffatt, H.K. 1978 Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, Cambridge.

    Google Scholar 

  • Ohkitani, K., Constantin, P. 2002, in preparation.

    Google Scholar 

  • SERRIN, J. 1959 Mathematical Principles of Classical Fluid Mechanics, In Handbuch der Physik, 8 (ed. S. Flugge and C. Truesdell), 125–263.

    Google Scholar 

  • Soward, A.M. 1972 A kinematic theory of large magnetic Reynolds number dynamos, Phil. Trans. Roy. Soc. A272, 431–462.

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics Chicago, The University of Chicago, Il, 60637, USA

    Peter Constantin

Authors
  1. Peter Constantin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Geophysics, Warsaw University, Warsaw, Poland

    K. Bajer

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

    H. K. Moffatt

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Kluwer Academic Publishers

About this paper

Cite this paper

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

Download citation

  • DOI: https://doi.org/10.1007/0-306-48420-X_35

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0980-8

  • Online ISBN: 978-0-306-48420-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation