On extensions of flows in the presence of sets of singularities
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.
KeywordsVector Field Velocity Field Measure Preserve Generate Vector Field Result Vector Field
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