Stable mixing estimates in the infinite Péclet number limit

Zelati, Michele Coti

doi:10.1016/j.jfa.2020.108562

Cited by 11 publications

(5 citation statements)

References 40 publications

(40 reference statements)

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“…As a final remark, the estimates given by the hypocoercivity method are naturally expressed in terms of the

H^1

‐type functional

\Phi

, and logarithmic corrections may appear when translating them into ordinary

L^2

estimates for the evolution equation (). It is worth noticing that the introduction of time‐dependent coefficients in

\Phi

can remove logarithmic losses [18, 22, 33, 43]. Moreover, the robustness of the method allows to treat different dissipative operators, for instance, fractional diffusion [33] or more complicated dissipative operators [8].…”

Section: Discussionmentioning

confidence: 99%

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Enhanced dissipation and Taylor dispersion in higher‐dimensional parallel shear flows

Gallay

Zelati

2023

Journal of London Math Soc

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We consider the evolution of a passive scalar advected by a parallel shear flow in an infinite cylinder with bounded cross section, in arbitrary space dimension. The essential parameters of the problem are the molecular diffusivity , which is assumed to be small, and the wave number in the streamwise direction, which can take arbitrary values. Under generic assumptions on the shear velocity , we obtain optimal decay estimates for large times, both in the enhanced dissipation regime and in the Taylor dispersion regime . Our results can be deduced from resolvent estimates using a quantitative version of the Gearhart–Prüss theorem, or can be established more directly via the hypocoercivity method. Both approaches are explored in the present example, and their relative efficiency is compared.

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“…As a final remark, the estimates given by the hypocoercivity method are naturally expressed in terms of the

H^1

‐type functional

\Phi

, and logarithmic corrections may appear when translating them into ordinary

L^2

estimates for the evolution equation (). It is worth noticing that the introduction of time‐dependent coefficients in

\Phi

Section: Discussionmentioning

confidence: 99%

“…\end{align}$$In fact, an even simpler proof can be carried out, without the use of the term multiplied by

\gamma

in (). See [16] for a proof in the one‐dimensional case.…”

Section: Energy Estimatesmentioning

confidence: 99%

Enhanced dissipation and Taylor dispersion in higher‐dimensional parallel shear flows

Gallay

Zelati

2023

Journal of London Math Soc

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“…We also refer to Refs. [1, 4, 20, 32, 33, 49, 72] for the enhanced dissipation in different models. Notations Through this paper, we will use the following notations.…”

Section: Introductionmentioning

confidence: 99%

A dynamical approach to the study of instability near Couette flow

Li,

Masmoudi,

Zhao

2023

Comm Pure Appl Math

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In this paper, we obtain the optimal instability threshold of the Couette flow for Navier–Stokes equations with small viscosity , when the perturbations are in the critical spaces . More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size with any small , which implies that is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo‐spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues.

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“…To the best of our knowledge, it is for the first time that lower bounds on mixing or enhanced dissipation are derived in the presence of diffusion. We also refer [6,14,12] and the literature therein for bounds and sharpness results in the case of shear flows.…”

Section: Introductionmentioning

confidence: 99%

“…The author likes to stress that ( 19) is derived under the assumption that the bounds derived in Theorem 2 and 3 are optimal. In view of the upper bound in (12), a version with a power α ∈ [1,2] on the logarithm should, however, be true.…”

Section: Introductionmentioning

confidence: 99%

Diffusion limited mixing rates in passive scalar advection

Seis¹

2020

Preprint

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We are concerned with flow enhanced mixing of passive scalars in the presence of diffusion. Under the assumption that the budget for the enstrophy is fixed during the evolution, we provide upper bounds on the exponential rates of mixing and of enhanced dissipation. Our results suggest that there is a crossover from advection dominated to diffusion dominated mixing, and we observe a slow down in the exponential decay rates by a logarithm of the diffusivity.

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Stable mixing estimates in the infinite Péclet number limit

Cited by 11 publications

References 40 publications

Enhanced dissipation and Taylor dispersion in higher‐dimensional parallel shear flows

Enhanced dissipation and Taylor dispersion in higher‐dimensional parallel shear flows

A dynamical approach to the study of instability near Couette flow

Diffusion limited mixing rates in passive scalar advection

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