Exponential self-similar mixing by incompressible flows

Abstract

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space W s,p , where s ≥ 0 and 1 ≤ p ≤ ∞. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev normḢ −1 , the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case s = 1 and 1 ≤ p ≤ ∞ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

“…There have been numerous recent contributions to this field: without any aim of completeness, we mention the works on lower bounds on mixing rates [30,39,43], mixing and regularity [15,31,52], and, more relevant for our discussion, quantification of mixing rates in passive scalars [1,10,22,53] (see also [2,4,38,44] and references therein for a dynamical system viewpoint) and two-dimensional Euler equations linearized around shear flows [27, 49-51, 54, 55] and vortices [6,19,56].…”

Universal mixers in all dimensions

“…2 ), as well as (A) in the affirmative for p ≥ 3+ and the geometric mixing scale only. (Additionally, [2] proves exponential mixing for all p ≥ 1 and some special initial data ρ(·, 0).) In this paper we answer…”

“…To see the latter, we note that (30) in [26] shows for mean-zero functions equivalence of theḢ −1/2 -norm and the mix-norm 3. 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.…”

On the Littlewood–Paley Spectrum for Passive Scalar Transport Equations

“…In the past years, fluid mixing attracted a remarkable attention by the mathematical fluid dynamics communities and beyond. The majority of the rigorous works, however, addressed the purely advective model, for instance, with a focus on absolute lower bounds on mixing rates [8,17,18,24,13], optimal mixing strategies [17,18,1,2,28], or universal mixers [11]. In the diffusive setting, it was shown that mixing flows enhance diffusive relaxation [6,4,7], while diffusion itself slows down the mixing rates [19].…”