Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations

Robinson, James C.; Sharples, Nicholas

doi:10.1016/j.jde.2013.02.012

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…Proof By (14) and since I ⊂ R 0 , it suffices to check that F i (R 0 ) ⊂ R 0 for i = 1, 2. Indeed, from this it will follow that Fw(R 0 ) ⊂ R 0 for all non-empty finite words w ∈ {1, 2} * .…”

Section: The Dimension Of Sαmentioning

confidence: 99%

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Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

Fefferman

Pooley

Rodrigo

2021

Arch Rational Mech Anal

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Section: The Dimension Of Sαmentioning

confidence: 99%

“…al. [14] using the notion of r-dimensional prints. [16], [17], and [13], in which such sufficient conditions for a flow to avoid a subset of [0, ∞) × R 3 are combined with partial regularity results for the Navier-Stokes equations to yield uniqueness of almost every trajectory for suitable weak solutions.…”

Section: Introductionmentioning

confidence: 99%

Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

Fefferman

Pooley

Rodrigo

2021

Arch Rational Mech Anal

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“…Since U is uniformly Lipschitz, its trajectory map X U is well defined and continuous, hence if we can show that X U (1, F 0 w (I)) = γF 0 w (I) for all w ∈ {1, 2} * then by (14)…”

Section: Contraction Of S α Due To Umentioning

confidence: 99%

Non-conservation of dimension in divergence-free solutions of passive and active scalar systems

Fefferman,

Pooley,

Rodrigo

2019

Preprint

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For any h ∈ (1, 2], we give an explicit construction of a compactly supported, uniformly continuous, and (weakly) divergence-free velocity field in R 2 that weakly advects a measure whose support is initially the origin but for positive times has Hausdorff dimension h.These velocities are uniformly continuous in space-time and compactly supported, locally Lipschitz except at one point and satisfy the conditions for the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di Perna and Lions theory.We then construct active scalar systems in R 2 and R 3 with measurevalued solutions whose initial support has co-dimension 2 but such that at positive times it only has co-dimension 1. The associated velocities are divergence free, compactly supported, continuous, and sufficiently regular to admit unique Regular Lagrangian Flows. This is in part motivated by the investigation of dimension conservation for the support of measure-valued solutions to active scalar systems. This question occurs in the study of vortex filaments in the three-dimensional Euler equations.

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Dimension prints for continuous functions on the unit square

et al. 2021

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Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations

Cited by 5 publications

References 12 publications

Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

Non-conservation of dimension in divergence-free solutions of passive and active scalar systems

Dimension prints for continuous functions on the unit square

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