On maximally mixed equilibria of two-dimensional perfect fluids

This paper advances the theory of maximally mixed equilibria in two-dimensional perfect fluids by demonstrating that minimizers of strictly convex Casimirs under fixed energy constraints are maximally mixed states, while also showing that on symmetric domains, certain initial data near shear or radial flows fail to weakly converge to these equilibria despite satisfying all conservation laws.

The Gist

Imagine a giant, invisible bathtub filled with a perfect, frictionless fluid (like an idealized version of water or air). If you swirl a spoon through it, creating a whirlpool, that swirl is called vorticity.

In the real world, friction eventually stops the swirl, and the water settles down. But in this "perfect" mathematical world, there is no friction. The swirls never die; they just get stretched, twisted, and folded like dough by the fluid's own motion.

The big question this paper asks is: If we wait forever, what does this swirling fluid look like? Does it eventually settle into a calm, predictable pattern, or does it keep churning in a chaotic mess forever?

Here is the breakdown of the paper's findings using simple analogies:

1. The "Shredder" Effect (Mixing)

Imagine you have a piece of paper with a drawing on it. If you put it through a paper shredder, the pieces get smaller and smaller. Eventually, you can't see the drawing anymore; you just have a pile of confetti.

In the fluid, the "shredder" is the flow itself. Over time, the initial swirls get shredded into tiny, fine filaments. If you look at the fluid with a "coarse" eye (ignoring the tiny details), it looks like the swirls have mixed together into a smooth, uniform gray. This is called mixing.

2. The Rules of the Game (Conservation Laws)

Even though the fluid is mixing, it has to follow strict rules, like a game with fixed constraints:

• Energy is Constant: The total "oomph" or speed of the fluid never changes. You can't create energy out of thin air.
• The "Shape" of the Swirl: The fluid can stretch and twist, but it can't destroy or create new "amounts" of swirl. It's like having a fixed amount of playdough; you can flatten it into a pancake or roll it into a snake, but you can't make it disappear.

3. The "Maximally Mixed" State

The authors are looking for the ultimate end state: the Maximally Mixed Equilibrium.

Think of it like this: Imagine you have a jar of red and blue marbles. You shake the jar.

• State A: The red marbles are all on top, blue on the bottom. (Not mixed).
• State B: They are perfectly blended into a purple soup. (Maximally mixed).

The paper proves that if the fluid reaches a state where it is impossible to mix it any further without breaking the rules (specifically, without changing the total energy), it must be a steady state. In other words, if the fluid is "as mixed as it can possibly be," it stops changing its large-scale appearance. It becomes a calm, steady flow.

The Big Surprise: The authors show that you can find these "perfectly mixed" states by solving a specific math puzzle (minimizing a "messiness" score). If you solve that puzzle, you get a flow that is stable and won't change.

4. The "Shear Flow" Trap

A Shear Flow is a very simple, orderly pattern where the fluid moves in parallel layers, like cards sliding over each other. In many physics theories, scientists thought that if you started with a messy swirl, the fluid would eventually "relax" and turn into a neat Shear Flow.

The paper says: "Not so fast!"

The authors constructed a specific, tricky starting point. Imagine a calm river (the Shear Flow) with two tiny, incredibly intense whirlpools hidden inside it.

• These whirlpools are so small and intense that they hold a massive amount of energy (like a compressed spring).
• The authors proved that it is impossible for the fluid to rearrange itself into a neat Shear Flow while keeping that energy constant.
• The "spring" is too tight. To turn into a Shear Flow, the fluid would have to lose that extra energy, but the rules say energy is conserved.

The Result: If you start with this specific messy setup, the fluid will never settle down into a neat, calm river. It will keep churning, perhaps forming complex, time-dependent patterns, but it will never become a simple Shear Flow.

5. Why This Matters

For a long time, people hoped that 2D fluids (like weather patterns or ocean currents) would eventually calm down into simple, predictable shapes. This paper says:

1. Yes, they can calm down if they reach a "maximally mixed" state (which looks like a steady flow).
2. But, they might not. If the initial conditions are "tricky" enough (like our hidden whirlpools), the fluid is kinematically forbidden from ever becoming a simple, calm flow. It is trapped in a state of perpetual, complex motion.

The Takeaway

Think of the fluid as a dancer.

• Old Theory: The dancer starts spinning wildly but will eventually slow down and stand perfectly still in a neat pose (Shear Flow).
• This Paper: The dancer can stop spinning and stand still, but only if they reach a specific "perfectly balanced" pose. However, if the dancer starts with a specific, tricky move, they are physically unable to ever stop and stand still. They are forced to keep dancing in a complex, never-ending routine forever.

The paper gives us the mathematical tools to know exactly when the dancer can stop, and when they are doomed to dance forever.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the fundamental problem of predicting the long-time behavior of two-dimensional (2D) perfect (inviscid and incompressible) fluids governed by the Euler equations.

• The Core Issue: While the Euler equations are time-reversible and conserve kinetic energy ($E$) and an infinite family of Casimir invariants ($I_f(\omega) = \int f(\omega) dx$), numerical simulations and physical intuition suggest that vorticity $\omega$ undergoes irreversible "mixing" over time. This leads to the formation of coherent large-scale structures (inverse energy cascade) while fine-scale filaments are lost.
• The Mathematical Challenge: In the limit $t \to \infty$, solutions may not converge strongly but rather converge in the weak-$*$ topology to a "coarse-grained" state. The set of all possible weak-$*$ limit points is the $\Omega$-limit set, $\Omega^+(\omega_0)$.
• Key Question: Can we characterize the possible end states of the fluid motion based solely on the conservation of energy and the transport structure of vorticity? Specifically, do "maximally mixed" states exist, and are they necessarily steady states? Furthermore, can we rule out convergence to specific symmetric equilibria (like shear flows) for certain initial data?

2. Methodology

The authors employ a combination of functional analysis, variational calculus, and dynamical systems theory:

• Phase Space: They work in the space $X = \{\omega \in L^\infty(M) : \|\omega\|_{L^\infty} \leq 1\}$ equipped with the weak-$*$ topology, which is compact and invariant under the Euler flow (Yudovich theory).
• Orbit Closures: They analyze the weak-$*$ closure of the orbit of the initial vorticity $\omega_0$ under area-preserving diffeomorphisms, denoted $\mathcal{O}_{\omega_0}^*$. This set represents all possible rearrangements of $\omega_0$ achievable through mixing.
• Variational Principles: The authors introduce a "mixing order" based on strictly convex Casimir functionals. They define $f$-minimal flows as elements in $\mathcal{O}_{\omega_0}^* \cap \{E=E_0\}$ that minimize a strictly convex functional $I_f$.
• Bistochastic Operators: They utilize the characterization of $\mathcal{O}_{\omega_0}^*$ via bistochastic operators (polymorphisms), linking the problem to the theory of rearrangement inequalities (Hardy-Littlewood-Polya) and Shnirelman's maximal mixing theory.
• Perturbation Analysis: To address convergence to shear flows, they construct specific initial data involving high-frequency, small-scale perturbations (regularized point vortices) and analyze the energy constraints required to rearrange them into shear flows.

3. Key Contributions and Results

A. Characterization of Maximally Mixed States (Theorem 1)

The paper provides a new perspective on A. Shnirelman's theory of maximally mixed states.

• Existence of Minimizers: For any strictly convex function $f$, there exists a minimizer $\omega^*$ of $I_f$ within the set of rearrangements with fixed energy $\mathcal{O}_{\omega_0}^* \cap \{E=E_0\}$.
• Equivalence to Shnirelman's Minimal Flows: The authors prove that any such $f$-minimizer is also a "minimal flow" in the sense of Shnirelman (defined via a preorder on the orbit closure).
• Stationarity and Structure: Crucially, they prove that any such minimizer $\omega^*$ is a stationary solution of the Euler equations. Furthermore, it satisfies a functional relation $\omega^* = F(\psi^*)$, where $\psi^*$ is the stream function and $F$ is a bounded monotone function.
• Variational Characterization: These minimizers are shown to be unconstrained minimizers of a specific functional involving the Casimir, energy, and a convex potential $\Phi$. This connects the problem to classical statistical hydrodynamics theories (like Miller-Robert-Sommeria).

B. Non-Convergence to Symmetric Equilibria (Theorem 2)

The paper challenges the assumption that 2D turbulence always relaxes to symmetric states (like shear flows in channels or circular flows in disks).

• Construction of Counter-Examples: The authors construct open sets of initial data (perturbations of any background shear flow $\omega_b$) consisting of highly peaked vortices embedded in the background.
• Energy Barrier: They demonstrate that for sufficiently small perturbation scales $\epsilon$, the energy of these perturbations scales as $|\log \epsilon|$.
• Impossibility of Rearrangement: They prove that it is impossible to rearrange these specific initial data into a shear flow (or radial flow) while conserving both energy and linear (or angular) momentum. The energy required to "smear out" the singular vortices into a 1D shear profile is significantly lower than the energy of the initial peaked state.
• Conclusion: Consequently, the Euler solution starting from such data cannot weakly converge to a shear flow. This implies that the long-time limit set $\Omega^+(\omega_0)$ does not contain any shear flows for these initial conditions.

C. Connection to Statistical Hydrodynamics

The paper bridges the gap between dynamical systems and statistical mechanics:

• It clarifies that Shnirelman's minimal flows are a subset of the predictions made by statistical hydrodynamics theories (which maximize entropy subject to constraints).
• It highlights that while statistical theories often predict convergence to specific stationary states (e.g., solutions to the Liouville or sinh-Poisson equations), the dynamical constraints (specifically the orbit closure $\mathcal{O}_{\omega_0}^*$) may restrict the accessible states, making some statistical predictions dynamically inaccessible.
• The authors discuss the role of Young measures in describing the fine-scale structure of the vorticity, showing that the coarse-grained limit $\bar{\omega}$ is a stationary solution, but the full distribution may retain memory of the initial data in a way that prevents convergence to simple symmetric states.

4. Significance and Implications

• Refutation of Universal Relaxation: The results provide a rigorous obstruction to the idea that 2D perfect fluids always relax to simple symmetric equilibria (like shear flows) via inviscid damping. The conservation of energy combined with the specific structure of the vorticity distribution can prevent this relaxation.
• Variational Framework for Equilibria: The paper establishes a robust variational framework for identifying possible long-time equilibria. It proves that "maximally mixed" states are not just abstract concepts but are concrete stationary solutions characterized by monotonic relations between vorticity and stream function.
• Bridging Theory and Simulation: The findings align with numerical simulations (Figure 1 and 3) where vortices persist and do not simply merge into a smooth shear flow, suggesting that the long-time dynamics may involve recurrent or time-dependent states rather than simple static equilibria.
• Open Questions: The paper raises important open questions regarding the uniqueness of these minimal flows, the regularity of the function $F(\psi)$, and whether the $\Omega$-limit set can ever be the entire set of rearrangements (ergodicity).

In summary, Dolce and Drivas provide a rigorous mathematical foundation for understanding the "end states" of 2D turbulence, proving that maximally mixed states are stationary equilibria and demonstrating that energy conservation can strictly forbid convergence to symmetric shear flows for a wide class of initial data.