Mathematical Problems in Theoretical Physics pp 405-414 | Cite as

# On extensions of flows in the presence of sets of singularities

## Abstract

Measure preserving (m.p.), flows in ℝ^{n} 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 ℝ^{n} n >
3, these phenomena can not be ruled out by the proposed condition, vεL^{2}, 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 vεL^{p} 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## Preview

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## References

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