Universal mixers in all dimensions

Elgindi, Tarek M.; Zlatoš, Andrej

doi:10.1016/j.aim.2019.106807

Cited by 42 publications

(38 citation statements)

References 29 publications

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“…Another interesting case to consider is when the advecting velocity field u is not bounded in W 1,∞ REMARK 4.5. The constant c 0 is independent of r and for the specific velocity field constructed in [26], we have that the velocity field is relaxation enhancing on the time scale ν −0.62 . We are also not aware of any velocity field on T 2 , other than the one constructed in [26], which causes all L 2 mean-free solutions of the advection diffusion equation to dissipate at a time scale of ν −q for any q < 1.…”

Section: Non-smooth Passive Scalarsmentioning

confidence: 93%

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

Zelati

Delgadino

Elgindi

2019

Comm Pure Appl Math

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We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect that causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a timescale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation timescales. The proofs are based on a contradiction argument that takes advantage of the cascading mechanism due to mixing, an estimate of the distance between the inviscid and viscous dynamics, and an optimization step in the frequency cutoff.Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation timescales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation timescale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion.

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Section: Non-smooth Passive Scalarsmentioning

confidence: 93%

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

Zelati

Delgadino

Elgindi

2019

Comm Pure Appl Math

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“…In [1], we constructed examples of velocity fields satisfying the required W 1,p bound and solutions to the associated continuity equation that saturate the exponential rate of decay of the negative norms (see [12,21] for related examples). We will refer to these examples as "optimal mixers".…”

Section: An Example Of Exponential Mixingmentioning

confidence: 99%

“…One can attempt to remedy this problem in two ways and obtain a stronger version of Theorem 4, by constructing exponential mixers with uniform bounds on higher derivatives or by constructing flows that give directly growth of positive norms for the solution. Very recently, Elgindi and Zlatos have constructed more general exponential mixers under uniform bounds in W s,p for 1 < s < 2 and p close to 2 [12]. (See Theorem 6 for more details on the examples of optimal mixers from [1].)…”

Section: Introductionmentioning

confidence: 99%

Loss of Regularity for the Continuity Equation with Non-Lipschitz Velocity Field

2019

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We consider the Cauchy problem for the continuity equation in space dimension d ≥ 2. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces W 1,p , for 1 ≤ p < ∞, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in W r,p , with r > 1, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space W r,p does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper Exponential self-similar mixing by incompressible flows (J. Amer. Math. Soc., DOI:https://doi.org/10.1090/jams/913), and have been announced in Exponential self-similar mixing and loss of regularity for continuity equations (C.

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“…Observe that the example in (1.2) is clearly not of cellular type. Examples of exponential universal mixers of non cellular type have been constructed in [11].…”

Section: Introductionmentioning

confidence: 99%

Polynomial mixing under a certain stationary Euler flow

Crippa

Lucà

Schulze

2019

Physica D: Nonlinear Phenomena

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We study the mixing properties of a scalar ρ on the unit disk advected by a certain incompressible velocity field u, which is a stationary radial solution of the Euler equation. The scalar ρ solves the continuity equation with the velocity field u and we can measure the degree of "mixedness" of ρ with two different scales commonly used in this setting, namely the geometric and the functional mixing scale. We develop a physical space approach well adapted to the quantitative analysis of the decay in time of the geometric mixing scale, which turns out to be polynomial for a large class of initial data. This extends previous results for the functional mixing scale, based on the explicit expression for the solution in Fourier variable, results that are also partially recovered by our approach.

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Universal mixers in all dimensions

Cited by 42 publications

References 29 publications

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

Loss of Regularity for the Continuity Equation with Non-Lipschitz Velocity Field

Polynomial mixing under a certain stationary Euler flow

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