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On extensions of flows in the presence of sets of singularities

  • Michael Aizenman
Short Communications
Part of the Lecture Notes in Physics book series (LNP, volume 80)

Abstract

Measure preserving (m.p.), flows in ℝn may have intersecting trajectories. This kind of singularity may prevent the uniqueness or, in other cases, even the existence of a m.p. flow with a given velocity field v, without contradicting the “integrability condition”:Δv = 0 in a weak sense. Examples are constructed to prove that, in ℝn n > 3, these phenomena can not be ruled out by the proposed condition, vεL2, or by any condition on the moments of v. Thus the question of a useful criterion, especially for the study of non stationery flows (e.g. those described by Navier-Stokes equations) remains open. On the positive side, a criterion is given which ensures that a measure preserving flow, with vεLp p > 1, has no flux through compact sets, e.g. sets of possible singularities, whose “dimension” is low enough. (The “dimension”, as used here, is not necessarily integral.) The upper bound on the “dimension” increases with p via an inequality which is optimal (possibly not strictly) as shown by the examples.

Keywords

Vector Field Velocity Field Measure Preserve Generate Vector Field Result Vector Field 
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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References

  1. [1]
    E. Nelson, Topics in Dynamics-I: Flows, Princeton University Press (1969).Google Scholar
  2. [2]
    E. Nelson, “Les ecoulements incompressibles d'energie finie”, Colloques Internationaux du Centre National de la Recherche Scientifique, 117 p. 159, 1962.Google Scholar
  3. [3]
    L.S. Young, “Entropy of continuous flows on compact 2-manifolds” (preprint).Google Scholar
  4. [4]
    M. Aizenman, “On vector fields as generators of flows; a counterexample to Nelson's conjecture,” Annals of Mathematics (to appear).Google Scholar
  5. [5]
    H. Federer, Geometric Measure Theory, Springer-Verlag (1969).Google Scholar
  6. [6]
    M. Aizenman, “A sufficient condition for the avoidance of sets by measure preserving flows in ℝn” (preprint).Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Michael Aizenman
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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