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arxiv: 2204.03587 · v2 · submitted 2022-04-07 · 🧮 math.AP · math-ph· math.MP· physics.flu-dyn

On maximally mixed equilibria of two-dimensional perfect fluids

Michele Dolce , Theodore D. Drivas This is my paper

Pith reviewed 2026-05-24 11:30 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPphysics.flu-dyn
keywords vorticity transportCasimir functionalsmaximal mixingperfect fluidsweak convergencestatistical hydrodynamicsarea-preserving diffeomorphismsequilibria
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The pith

Any minimizer of a strictly convex Casimir over fixed-energy vorticity states is a maximally mixed equilibrium.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Two-dimensional perfect fluids transport vorticity by area-preserving flows while keeping kinetic energy fixed. Possible long-time states therefore occupy the weak closure of the initial orbit intersected with the energy surface. The paper proves that any such state minimizing a strictly convex Casimir must be maximally mixed, so that no further mixing is possible without changing the energy. This shows that vorticity transport and energy conservation by themselves cannot rule out weak convergence to equilibrium. On symmetric domains the same set of constraints yields open families of initial data that remain close in L1 to shear or radial flows yet fail to converge weakly to them.

Core claim

Any minimizer of any strictly convex Casimir in the set of vorticity fields with fixed energy, denoted by the weak closure of the orbit under area-preserving diffeomorphisms intersected with the energy constraint, is maximally mixed. Thus weak convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On domains with symmetry the same set admits open families of initial data arbitrarily close in L1 to shear or radial flows that nevertheless do not converge weakly to those flows.

What carries the argument

The set of possible end states given by the weak closure of the vorticity orbit under area-preserving diffeomorphisms intersected with the fixed-energy surface, on which strictly convex Casimirs attain their minima.

If this is right

  • Weak convergence to equilibrium remains possible under the constraints of ideal two-dimensional motion.
  • On domains with symmetry such as channels or annuli, open sets of initial data exist that stay arbitrarily close in L1 to shear or radial flows without weakly converging to them.
  • The variational characterization supplies a direct link between convex Casimir minimization and the notion of maximal mixing introduced by Shnirelman.
  • Classical statistical hydrodynamics theories receive a new selection principle for equilibria that are attainable by ideal evolution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Additional selection mechanisms beyond energy and Casimirs may be required to determine which equilibrium is actually reached.
  • Numerical experiments could check whether typical random initial data evolve toward Casimir minimizers on the fixed-energy set.
  • The same variational argument might apply to other transport problems that conserve a family of integral invariants together with one quadratic quantity.

Load-bearing premise

The weak closure of the orbit under area-preserving diffeomorphisms together with the fixed-energy constraint forms a setting in which strictly convex Casimirs attain their minima and maximal mixing is well-defined.

What would settle it

An explicit vorticity field inside the fixed-energy weak closure that minimizes a strictly convex Casimir yet still permits further mixing at constant energy would disprove the claim.

read the original abstract

The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution $\omega_0$, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing of $\omega_0$ can occur. Since kinetic energy $\mathsf{E}$ is conserved, not all the mixed states are relevant and it is natural to consider only the ones with energy $\mathsf{E}_0$ corresponding to $\omega_0$. The set of said vorticity fields, denoted by $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept of maximally mixed states (any further mixing would necessarily change their energy), and proved they are perfect fluid equilibria. We offer a new perspective on this theory by showing that any minimizer of any strictly convex Casimir in $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ is maximally mixed, as well as discuss its relation to classical statistical hydrodynamics theories. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, on domains with symmetry (e.g. straight channel or annulus), we exploit all the conserved quantities and the characterizations of $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ to give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow in $L^1$ of vorticity but do not weakly converge to them in the long time limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims that any minimizer of a strictly convex Casimir on the set S = weak closure of the orbit of initial vorticity under area-preserving diffeomorphisms, intersected with the fixed-energy level set, is a maximally mixed equilibrium (in the sense of Shnirelman). It concludes that weak convergence to equilibrium cannot be ruled out by vorticity transport and energy conservation alone, and provides examples on symmetric domains (channel, annulus) of open sets of initial data that stay L1-close to shear/radial flows but do not converge weakly to them.

Significance. If the central claims hold, the result links minimization of strictly convex Casimirs directly to maximal mixing and shows that the conserved quantities alone do not obstruct weak convergence to equilibrium. The symmetric-domain counterexamples supply concrete, falsifiable instances where long-time behavior deviates from naive expectations. The manuscript does not supply machine-checked proofs or parameter-free derivations.

major comments (2)
  1. [Abstract (paragraph beginning 'The set of said vorticity fields')] Abstract (paragraph beginning 'The set of said vorticity fields'): The central claim that minimizers of strictly convex Casimirs on S are maximally mixed presupposes that such minimizers exist. No argument is given that S is weakly compact, that the Casimir is weakly lower-semicontinuous on S, or that the fixed-energy constraint {E=E0} is weakly closed. Since kinetic energy is only weakly lower-semicontinuous, level sets need not be weakly closed; without uniform integrability or a-priori bounds on vorticity, existence can fail. This is load-bearing for both the main theorem and the conclusion that convergence cannot be excluded.
  2. [Abstract (final paragraph on symmetric domains)] Abstract (final paragraph on symmetric domains): The examples of open sets of initial data that remain L1-close to shear or radial flows but do not converge weakly rely on an explicit characterization of the weak closure and all conserved quantities. The manuscript supplies no verification that the constructed sets are indeed open in the appropriate topology or that the non-convergence persists under the full vorticity-transport dynamics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: Abstract (paragraph beginning 'The set of said vorticity fields'): The central claim that minimizers of strictly convex Casimirs on S are maximally mixed presupposes that such minimizers exist. No argument is given that S is weakly compact, that the Casimir is weakly lower-semicontinuous on S, or that the fixed-energy constraint {E=E0} is weakly closed. Since kinetic energy is only weakly lower-semicontinuous, level sets need not be weakly closed; without uniform integrability or a-priori bounds on vorticity, existence can fail. This is load-bearing for both the main theorem and the conclusion that convergence cannot be excluded.

    Authors: We agree the manuscript provides no proof of existence of minimizers. The main result is the characterization that any such minimizer (when it exists) must be maximally mixed; this statement is logically valid even if the set of minimizers is empty. For the conclusion that weak convergence cannot be ruled out by the conserved quantities alone, existence would indeed be needed to exhibit concrete equilibria reachable as weak limits. We will revise the abstract and introduction to state explicitly that existence is not established and the result is conditional on the existence of minimizers. We do not claim general weak compactness or lower semicontinuity without additional assumptions on vorticity. revision: partial

  2. Referee: Abstract (final paragraph on symmetric domains): The examples of open sets of initial data that remain L1-close to shear or radial flows but do not converge weakly rely on an explicit characterization of the weak closure and all conserved quantities. The manuscript supplies no verification that the constructed sets are indeed open in the appropriate topology or that the non-convergence persists under the full vorticity-transport dynamics.

    Authors: The constructions in the symmetric-domain sections use the explicit characterization of the weak closure together with the full list of conserved quantities preserved by the symmetry (e.g., all moments along the channel or radial moments in the annulus). The sets are defined by open conditions on these invariants that exclude the target shear/radial flow as a weak limit while keeping the data L1-close; openness follows directly from the continuity of the conserved quantities in L1. Non-convergence is shown by verifying that every possible weak limit point must satisfy an additional constraint violated by the equilibrium. We will add a short clarifying lemma summarizing these openness and invariance arguments if the referee finds the current presentation insufficient. revision: no

Circularity Check

0 steps flagged

No circularity: result follows from transport, energy conservation, and convex functional properties

full rationale

The paper defines the set S = weak closure of the orbit under area-preserving diffeomorphisms intersected with fixed energy, then proves that any minimizer of a strictly convex Casimir on S is maximally mixed. This is a direct consequence of the definitions of mixing, the transport equation, and strict convexity; no step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. Shnirelman's prior result on equilibria is cited as background but is not used to force the new claim. The derivation remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard functional-analytic properties of the 2D Euler vorticity equation and the definition of weak orbit closures; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The weak closure of the orbit under area-preserving flows together with the fixed-energy slice forms a suitable space in which Casimirs are well-defined and attain minima.
    Invoked when the set of possible end states is introduced in the abstract.
  • standard math Strict convexity of the Casimir functionals is preserved under the relevant weak limits.
    Used to conclude that minimizers are maximally mixed.

pith-pipeline@v0.9.0 · 5868 in / 1394 out tokens · 41186 ms · 2026-05-24T11:30:56.556838+00:00 · methodology

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