On the Littlewood–Paley Spectrum for Passive Scalar Transport Equations

Seis, Christian

doi:10.1007/s00332-019-09585-w

Cited by 3 publications

(4 citation statements)

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“…Case (2) is compatible with the Batchelor scaling of the spectrum (1.1). In [22], a modified LP spectrum, which uses the L1 norm of Δjθκ, instead of the L2 norm was studied. An upper bound of order k−1 for its decay in a suitable weighted long-time average was obtained, assuming the velocity field u is in the Sobolev space Hs, s∈false[0,1false], uniformly in time.…”

Section: The Scalar Spectrum and Scalar Dissipationmentioning

confidence: 99%

“…All these results concern pure mixing and transport of the scalar field and can be considered in the limit of infinite Schmidt number. There are significantly fewer works in pde/analysis addressing mixing with diffusion and, in particular, bounds on the energy dissipation rate or the spectrum for scalar turbulence at large, but finite, Schmidt number [21][22][23][24]. In fact, transport can both enhance as well as balance diffusion.…”

Section: Introductionmentioning

confidence: 99%

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Remarks on anomalous dissipation for passive scalars

Mazzucato

2022

Phil. Trans. R. Soc. A.

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We consider the problem of anomalous dissipation for passive scalars advected by an incompressible flow. We review known results on anomalous dissipation from the point of view of the analysis of partial differential equations, and present simple rigorous examples of scalars that admit a Batchelor-type energy spectrum and exhibit anomalous dissipation in the limit of zero scalar diffusivity. This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.

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Section: The Scalar Spectrum and Scalar Dissipationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Remarks on anomalous dissipation for passive scalars

Mazzucato

2022

Phil. Trans. R. Soc. A.

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“…In view of the fact that a mollification in physical space θ * φ ǫ corresponds to a cut-off in Fourier space θ * φ k , our approach to regularity in Theorem 1 is a sharp quantification of DiPerna and Lions's commutator lemma. A crude estimate on the Littlewood-Paley commutator was used earlier by one of the authors to estimate the L 1 -based energy spectrum in the diffusive setting [35].…”

Section: Introductionmentioning

confidence: 99%

Propagation of regularity for transport equations. A Littlewood-Paley approach

Meyer¹,

Seis²

2022

Preprint

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It is known that linear advection equations with Sobolev velocity fields have very poor regularity properties: Solutions propagate only derivatives of logarithmic order, which can be measured in terms of suitable Gagliardo seminorms. We propose a new approach to the study of regularity that is based on Littlewood-Paley theory, thus measuring regularity in terms of Besov norms. We recover the results that are available in the literature and extend these optimally to the diffusive setting. As a consequence, we derive sharp bounds on rates of convergence in the zero-diffusivity limit.

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“…[36,37,40,69,84,120]); estimation of possible power spectra (see e.g. [38,41,111,119]); characterization of sufficient conditions for predictions relating to turbulence (see e.g. [34,46,52,55,107]); and recent progress on wave turbulence [31,32].…”

Section: Introductionmentioning

confidence: 99%

The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics at fixed Reynolds number

Bedrossian¹,

Blumenthal²,

Punshon-Smith³

2019

Preprint

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In 1959, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity κ should display a |k| −1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimentally and numerically tested extensively in the physics and engineering literature and is a core prediction of passive scalar turbulence.In this article we provide the first mathematically rigorous proof of Batchelor's prediction on the cumulative power spectrum in the κ → 0 limit. We consider fluids governed by the 2D Navier-Stokes equations and 3D hyperviscous Navier-Stokes equations in T d forced by sufficiently regular, nondegenerate stochastic forcing at fixed (arbitrary) Reynolds number. The scalar is subjected to a smooth-inspace, white-in-time stochastic source, and evolves by advection-diffusion with diffusivity κ > 0. Our results rely on the quantitative understanding of Lagrangian chaos and passive scalar mixing established in our recent works. In the κ → 0 limit, we obtain statistically stationary, weak solutions in H − to the stochastically-forced advection problem without diffusivity. These solutions are almost-surely not locally integrable functions, have a non-vanishing average anomalous flux, and satisfy the Batchelor spectrum at all sufficiently small scales. We also prove an Onsager-type criticality result which shows that no such dissipative, weak solutions with a little more regularity can exist.

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On the Littlewood–Paley Spectrum for Passive Scalar Transport Equations

Abstract: We derive time-averaged L 1 estimates on Littlewood-Paley decompositions for linear advection-diffusion equations. For wave numbers close to the dissipative cut-off, these estimates are consistent with Batchelor's predictions on the variance spectrum in passive scalar turbulent mixing.

Cited by 3 publications

References 27 publications

Remarks on anomalous dissipation for passive scalars

Remarks on anomalous dissipation for passive scalars

Propagation of regularity for transport equations. A Littlewood-Paley approach

The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics at fixed Reynolds number

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