Dissipative solutions to randomly forced 3D Euler equations

Imagine you are trying to predict the weather, or how a drop of ink spreads in a glass of water. Scientists use complex mathematical equations (like the Euler equations) to describe how fluids move. For a long time, mathematicians believed that if you knew the starting position of the fluid perfectly, you could predict exactly how it would move forever. It was like a giant, perfect clockwork machine.

However, this paper by Umberto Pappalettera and Francesco Triggiano shows that fluids are much more chaotic and unpredictable than we thought, especially when you add a little bit of "randomness" (like tiny, invisible bumps from heat or environmental noise).

Here is the breakdown of their discovery using simple analogies:

1. The Problem: The "Perfect" Fluid vs. The "Messy" Reality

Think of a fluid as a crowd of people dancing.

The Old View: If you knew exactly where everyone was and how fast they were moving, you could predict the dance steps for the next hour. The dance would be smooth and orderly.
The New View: The authors show that even if you know the starting positions, the dance can suddenly become a chaotic mosh pit. There isn't just one way the dance can evolve; there are many different ways it could happen, and they are all mathematically valid.

2. The Experiment: Adding "Static" to the Signal

In this paper, the authors didn't just look at a calm fluid; they added random noise (like static on an old TV or a gentle, unpredictable breeze).

The Analogy: Imagine you are trying to balance a broom on your hand. If you just stand still, it's hard. But if someone keeps gently poking you from random directions (the "noise"), the broom starts wobbling in ways that are impossible to predict precisely.
The Discovery: They proved that with this random poking, you can construct multiple different futures for the fluid. It's not just that the future is hard to calculate; it's that the math allows for multiple, equally valid realities to exist simultaneously.

3. The "Dissipative" Secret: The Fluid Loses Energy

One of the most important parts of their work is the concept of "Dissipative Solutions."

The Analogy: Think of a spinning top. In a perfect, frictionless world, it would spin forever. In the real world, it slows down and stops because it loses energy to friction (heat).
The Math: The authors built solutions where the fluid loses energy on purpose, even though the equations say it shouldn't. They showed that the fluid can create tiny, invisible "turbulence" (like microscopic whirlpools) that suck energy out of the system.
Why it matters: This explains why real fluids (like air or water) eventually calm down or behave differently than the "perfect" math models predict. The fluid is "dissipating" energy into these tiny, chaotic swirls.

4. The "Ergodicity" Puzzle: Two Different Histories

The second part of the paper tackles a deep question in physics called Ergodicity.

The Question: If you watch a fluid for a very long time, does it eventually settle into a single, average "personality"? Or can it have different "personalities" depending on how it started?
The Analogy: Imagine two identical cars driving on the same road with the same engine.
- Car A drives smoothly and ends up at a specific average speed.
- Car B drives erratically and ends up at a different average speed.
- The authors proved that for these randomly forced fluids, both cars are valid. You cannot predict which "personality" the fluid will have just by looking at the road (the external force). The fluid has a "memory" or a "choice" that leads to different statistical outcomes.

5. How Did They Do It? (The "Convex Integration" Trick)

How do you prove something that seems impossible? The authors used a technique called Convex Integration.

The Analogy: Imagine you are trying to build a sculpture out of clay, but you are only allowed to use very specific, tiny shapes.
1. You start with a rough shape.
2. You realize it's not quite right, so you add a layer of tiny, wiggly bumps to fix the errors.
3. Those bumps create new tiny errors, so you add an even finer layer of bumps to fix those.
4. You keep doing this, adding layers of "noise" that get smaller and smaller, until the sculpture looks smooth from a distance, but up close, it's a chaotic mess of tiny corrections.
The Result: By doing this mathematically, they constructed these "dissipative" solutions that satisfy all the rules but behave wildly.

Summary: What Does This Mean for Us?

This paper is a major breakthrough because it shatters the idea that fluid dynamics is a simple, predictable clockwork machine.

Unpredictability is built-in: Even with perfect knowledge of the start, the future of a fluid with random noise is not unique.
Energy loss is real: Fluids can naturally lose energy through these chaotic, microscopic swirls, which explains why real-world turbulence happens.
Multiple Realities: The same external force (like wind) can lead to two completely different long-term behaviors for a fluid.

In short, the universe of fluids is messier, more chaotic, and more full of possibilities than we previously believed. The math doesn't just describe the fluid; it reveals that the fluid has a hidden, chaotic soul that refuses to be pinned down to a single story.

Here is a detailed technical summary of the paper "Dissipative Solutions to Randomly Forced 3D Euler Equations" by Umberto Pappalettera and Francesco Triggiano.

1. Problem Statement and Context

The paper addresses two fundamental problems in the mathematical theory of fluid dynamics regarding the 3D Euler equations perturbed by additive noise (stochastic forcing):

Existence and Non-uniqueness of Dissipative Solutions: While the deterministic Euler equations ( $Q=0$ ) are known to admit non-unique weak solutions via convex integration (De Lellis, Székelyhidi, Isett, etc.), the stochastic case ( $Q \neq 0$ ) presents unique challenges. Specifically, the authors aim to construct probabilistically strong solutions that satisfy the local energy inequality (LEI) up to an arbitrarily large stopping time. This is crucial because the LEI is the standard criterion for "suitable" weak solutions (analogous to Caffarelli-Kohn-Nirenberg for Navier-Stokes), ensuring that energy is not artificially created by singularities.
Non-uniqueness of Ergodicity: The paper investigates the ergodic hypothesis for the forced Euler equations. It seeks to prove that the system admits multiple distinct ergodic invariant measures concentrated on solutions that are:
- Strictly dissipative (energy is lost locally).
- Genuinely random (not steady states).
- Not uniquely determined by the law of the external forcing.

2. Methodology

The core methodology relies on a sophisticated adaptation of the Convex Integration Scheme, originally developed for deterministic Euler equations, to the stochastic setting.

A. Decomposition and Reformulation

Da Prato-Debussche Trick: For the stochastic Euler equations (SE), the authors decompose the velocity field as $u = v + z$ , where $z = Q^{1/2}W$ is the stochastic convolution (the linear part driven by noise) and $v$ is the remainder. This transforms the stochastic PDE into a random PDE for $v$ with a modified nonlinearity.
Euler-Reynolds Flow: The authors construct approximate solutions $(v_q, p_q, R_q, \phi_q, z_q)$ satisfying a relaxed system where the momentum equation includes a Reynolds stress tensor $R_q$ and the local energy inequality includes a flux current $\phi_q$ . The goal is to drive these errors to zero as $q \to \infty$ .

B. Iterative Scheme

The construction proceeds via an iterative sequence $q \in \mathbb{N}$ :

Perturbation Construction: A velocity perturbation $w = v_{q+1} - v_q$ is constructed using Mikado flows (highly oscillating vector fields).
Amplitude Choice: The amplitudes of the Mikado flows are carefully chosen to cancel out the existing Reynolds stress $R_q$ and the flux current $\phi_q$ in the next iteration.
Stochastic Adaptation:
- The scheme handles the temporal roughness of the noise by introducing time mollification (asymmetric to preserve adaptedness to the filtration).
- Local Energy Inequality: A key innovation is the construction of a specific flux current $\phi$ and a scalar function $f(t)$ (energy loss) to ensure the local energy balance holds strictly.
- Stopping Time: The iteration is controlled up to a stopping time $\mathfrak{t}$ defined by the regularity of the noise $z$ , ensuring the solution remains valid with high probability.

C. Ergodicity Construction

To prove non-uniqueness of ergodic measures:

The authors construct two distinct sequences of solutions starting from different initial data (or different scaling of the noise interaction) but driven by the same law of external forcing.
They utilize the Krylov-Bogoliubov theorem to extract statistically stationary solutions from the time averages of the constructed flows.
By controlling the "dissipation" parameter $E'$ , they ensure the resulting measures are concentrated on strictly dissipative, non-steady states.

3. Key Contributions

1. Probabilistically Strong Dissipative Solutions (Theorem 1.1)

Result: The authors prove the existence of probabilistically strong solutions to the 3D stochastic Euler equations that are:
- Continuous in time and Hölder continuous in space ( $C^\alpha$ ).
- Satisfy the Local Energy Inequality (LEI) almost surely up to a stopping time $\mathfrak{t}$ .
- Strictly dissipative: The local energy inequality is strict (energy is dissipated) if the energy loss rate $E' > 0$ .
Significance: This extends the non-uniqueness results of Hofmanová, Zhu, and Zhu (2022) by establishing the local energy inequality, a much stronger physical condition than global energy bounds. It confirms that non-uniqueness persists even when restricting to physically "suitable" solutions.

2. Non-Unique Ergodicity (Theorem 1.2 & 1.4)

Result: The paper constructs two distinct ergodic measures $\mu_1$ $μ_{1}$ and $\mu_2$ $μ_{2}$ on the path space of the forced Euler equations.
- Both measures are concentrated on solutions satisfying the local energy inequality.
- Under both measures, the velocity field $u$ is genuinely random (non-zero variance) and not a steady state.
- Crucially, the law of the external forcing $g$ is identical under both measures, yet the laws of the velocity $u$ differ.
Significance: This resolves a major open problem regarding the ergodicity of the Euler equations. It demonstrates that the "vanishing viscosity limit" is insufficient to select a unique statistical state for the Euler equations. The system exhibits non-unique ergodicity, meaning the long-term statistical behavior depends on the specific solution trajectory, not just the forcing statistics.

3. Technical Innovations

Stochastic Convex Integration: The paper successfully adapts the complex geometric lemmas (Mikado flows, partition of unity) to the stochastic setting, managing the interplay between the noise regularity and the high-frequency oscillations required for convex integration.
Local Energy Control: The construction of the flux current $\phi$ and the scalar correction $f(t)$ allows for the enforcement of the local energy inequality in the stochastic context, a significant technical hurdle compared to global energy estimates.

4. Main Results Summary

Theorem	Statement	Implication
Theorem 1.1	Existence of probabilistically strong solutions to (SE) satisfying the Local Energy Inequality (LEI) up to a large stopping time.	Non-uniqueness holds even for "suitable" weak solutions in the stochastic setting.
Theorem 1.2	Existence of two distinct ergodic measures $\mu_1, \mu_2$ for the forced Euler equations (FE) with the same forcing law but different velocity laws.	The ergodic hypothesis fails; the system does not have a unique statistical equilibrium.
Theorem 1.4	Construction of ergodic measures where the velocity law can be arbitrarily prescribed (up to a small error) while maintaining a fixed dissipation rate.	Demonstrates extreme flexibility in the statistical behavior of the system; the map from forcing law to solution law is not unique.

5. Significance and Impact

Fluid Dynamics: The results deepen the understanding of turbulence in ideal fluids. They suggest that even with external forcing, the 3D Euler equations do not settle into a unique statistical state, challenging the foundations of statistical hydrodynamics.
Mathematical Physics: The work bridges the gap between deterministic convex integration (Onsager's conjecture) and stochastic analysis. It shows that the "wild" behavior of Euler solutions (non-uniqueness, energy dissipation) is robust against the addition of random noise.
Ergodic Theory: By proving non-unique ergodicity for the Euler equations, the paper provides a counterexample to the idea that adding noise and forcing automatically leads to a unique invariant measure (a property often hoped for in stochastic PDEs like Navier-Stokes).

In conclusion, Pappalettera and Triggiano have successfully extended the convex integration technique to the stochastic 3D Euler equations, proving the existence of strictly dissipative, non-unique solutions and establishing the failure of unique ergodicity for the forced system. This work represents a major step forward in the rigorous mathematical theory of turbulence.