Polynomial mixing under a certain stationary Euler flow

Abstract

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.

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

“…REMARK 1.5. The case α = 1, from the point of view of mixing rates, has been analyzed in detail in [22], using different methods.…”

“…Stability of such radial solution in the 2D Euler equations were recently addressed in [6,19,56]. From the mixing point of view in the passive scalar problem (5.1), the case α = 1 was recently studied in great detail in [22], where the decay of theḢ −1 and the geometric mixing scale was proven under the natural the natural condition orthogonality condition ∂Bρ f in dS ρ = 0,…”

“…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].…”

Stable mixing estimates in the infinite Péclet number limit

“…It is therefore not surprising that a way to quantify this is through the decay of a negative Sobolev norm, as in (1.12). In the context of passive scalars, this point of view was introduced in [31], and it is deeply connected with the regularity of transport equations [12,18,26,45], the quantification and lower bounds on mixing rates [1,6,19,25,32,36,46], and the inviscid damping in the two-dimensional Euler equations linearized around shear flows [8,17,23,40,42,43,[47][48][49].…”

On the Forced Euler and Navier–Stokes Equations: Linear Damping and Modified Scattering

“…• As a related problem damping, mixing and enhanced dissipation for passive scalar problems are an active area of research [ZDE18], [ACM14], [CS17], [CLS17], [Zil18].…”