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.
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References
E. Nelson, Topics in Dynamics-I: Flows, Princeton University Press (1969).
E. Nelson, “Les ecoulements incompressibles d'energie finie”, Colloques Internationaux du Centre National de la Recherche Scientifique, 117 p. 159, 1962.
L.S. Young, “Entropy of continuous flows on compact 2-manifolds” (preprint).
M. Aizenman, “On vector fields as generators of flows; a counterexample to Nelson's conjecture,” Annals of Mathematics (to appear).
H. Federer, Geometric Measure Theory, Springer-Verlag (1969).
M. Aizenman, “A sufficient condition for the avoidance of sets by measure preserving flows in ℝn” (preprint).
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© 1978 Springer-Verlag
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Aizenman, M. (1978). On extensions of flows in the presence of sets of singularities. In: Dell'Antonio, G., Doplicher, S., Jona-Lasinio, G. (eds) Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-08853-9_35
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DOI: https://doi.org/10.1007/3-540-08853-9_35
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