On the Relation between Enhanced Dissipation Timescales and Mixing Rates

Abstract

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.

Diffusion and mixing in fluid flow via the resolvent estimate

“…Recently, Coti Zelati, Delgadino and Elgindi proved the enhanced dissipation time-scale O(| ln ν| 2 ) in (1.2) for all contact Anosov flows on a smooth 2d + 1 dimensional connected compact Riemannian manifold [6]. Their proof is based on the knowledge of mixing decay rates.…”

Universal mixers in all dimensions

“…6. We also remark that it follows from (i) that mean-zero solutions to the corresponding advection-diffusion equation ρ t + u · ∇ρ = ν∆ρ with ν ∈ (0, 1 2 ) satisfy ρ(·, t) 2 ≤ e −cν 0.62 t ρ(·, 0) 2 for all t > ν −0.62 , with c a universal constant (see Theorem 4.4 in [11]). That is, their diffusive time scale is at most O(ν −0.62 ), which is much shorter than the time scale O(ν −1 ) for the heat equation (without advection) when 0 < ν 1.…”

“…OtherḢ −s -norms of f have been used to quantify mixing, particularly theḢ −1 -norm [2,21,23,25,31], with the functional mixing scale being f Ḣ−1 f −1 ∞ (Wasserstein distance of f + and f − has also been used [6,29,31,32]). The latter may sometimes be more convenient than theḢ −1/2 -norm and also is directly related to mixing-enhanced diffusion rates when diffusion is present [9,11], but it lacks the useful connection to the mix-norm. We use theḢ −1/2 -norm here but note that our mixing results for it also hold for theḢ −1 -norm because the former controls the latter.…”

On the Littlewood–Paley Spectrum for Passive Scalar Transport Equations

“…The second estimate simply follows from Young's convolution estimate and Jensen's inequality, For the first estimate, (19), we notice that in view of the scaling property (6), it is enough to establish the statement of ℓ = 0. Due to the dyadic partition of unity of the frequency space in (5)- (7), it holds that F φ −1 + F φ 0 + F φ 1 = 1 in the support of F φ 0 . As a consequence, φ −1 + φ 0 + φ 1 leaves φ 0,j invariant under convolution.…”