Nontrivial absolutely continuous part of anomalous dissipation measures in time

This paper resolves specific open questions regarding anomalous dissipation in four-dimensional incompressible Navier-Stokes equations by constructing new examples of solutions that exhibit a nontrivial absolutely continuous part in their dissipation measures and demonstrating the weak* convergence of these solutions to a smooth Euler limit as viscosity vanishes.

The Gist

Imagine you are watching a pot of water on the stove. As you turn up the heat, the water starts to swirl and churn. In physics, this is called turbulence.

For over 80 years, physicists have been puzzled by a specific mystery about this churning water, known as the "Zeroth Law of Turbulence."

Here is the simple version of the problem:

• Viscosity is like the "thickness" or "stickiness" of the fluid (like honey vs. water).
• In a real-world fluid, this stickiness causes energy to turn into heat (dissipation).
• The Zeroth Law suggests that even if you make the fluid perfectly smooth (removing all stickiness, or viscosity), the turbulence should still "eat" energy at a steady, continuous rate. It shouldn't just stop; it should keep dissipating energy smoothly over time.

However, mathematicians have struggled to prove this. Most examples they found were "tricks" where the energy dissipation happened in a sudden, jagged burst at the very end of the experiment, rather than flowing smoothly like a river.

The Big Breakthrough

This paper by Carl Johan Peter Johansson and Massimo Sorellia is like finding the "Holy Grail" of fluid dynamics. They have built a mathematical model (a virtual experiment) that finally proves: Yes, you can have turbulence that dissipates energy smoothly and continuously over time, even when the fluid has zero stickiness.

Here is how they did it, using some creative analogies:

1. The 4D "Shadow" Trick

The authors didn't just look at a 3D pot of water. They built a 4-dimensional model.

• The Analogy: Imagine a 3D shadow puppet show. The shadows on the wall (3D) look complex, but the real puppets moving behind the screen (4D) are doing something even more intricate.
• By working in this higher dimension, they could construct a "force" (like a hand stirring the pot) that is time-independent (it doesn't change its pattern) but creates a flow that behaves exactly as the Zeroth Law predicts.

2. The "Mixing" Velocity Field

To get the energy to dissipate smoothly, they needed a very specific way of stirring the fluid.

• The Analogy: Think of a baker kneading dough. If you just push it once, it doesn't mix well. But if you stretch it, fold it, and stretch it again in a very specific, chaotic pattern, the dough mixes perfectly.
• The authors created a "velocity field" (a map of how the fluid moves) that acts like this master baker. It stretches and folds the fluid in a way that creates tiny, tiny swirls. These swirls are so small that even a tiny amount of friction (viscosity) turns a massive amount of energy into heat.
• Crucially, they proved that as they made the friction smaller and smaller (approaching zero), the rate of energy loss didn't stop; it settled into a smooth, continuous curve.

3. The "Ghost" Energy

In their model, they tracked the "energy" of the fluid.

• The Analogy: Imagine you are watching a car drive down a highway. Usually, if you remove the engine (friction), the car coasts forever. But in their model, even with the engine off, the car is losing speed smoothly, as if it's driving through invisible, thick air.
• They proved that this "invisible air" (the anomalous dissipation) is real and has a smooth, continuous profile. It's not a sudden crash; it's a gentle, steady decline in energy, just like the Zeroth Law predicted.

Why This Matters

Before this paper, we had examples of turbulence, but they were "broken" examples where the energy loss happened in a weird, jagged spike at the very end.

This paper provides the first rigorous proof that:

1. You can have a fluid that is perfectly smooth (mathematically) but still loses energy.
2. This energy loss happens continuously over time, not in a sudden burst.
3. This behavior is "close" to what we see in real-world experiments and computer simulations of weather and ocean currents.

The Takeaway

Think of this paper as finally finding the missing piece of a giant puzzle. For decades, we knew the picture should look a certain way (smooth energy loss), but we couldn't build the pieces to fit. Johansson and Sorellia built a new set of pieces in a 4-dimensional world that fit perfectly, proving that the "Zeroth Law" is mathematically possible.

They didn't just say "it's possible"; they built the exact blueprint and showed that the energy dissipation is a smooth, flowing river, not a jagged waterfall. This brings us one step closer to truly understanding the chaotic beauty of turbulence in our universe.

Technical Summary

1. Problem Statement and Context

The paper addresses fundamental questions regarding anomalous dissipation in fluid dynamics, specifically within the context of the zeroth law of turbulence (Kolmogorov's 1941 theory). The central issue is whether the energy dissipation rate remains non-zero in the limit of vanishing viscosity ($\nu \to 0$) for solutions to the incompressible Navier-Stokes equations, even when the limiting solution satisfies the inviscid Euler equations.

Key Definitions and Challenges:

• Anomalous Dissipation: The phenomenon where $\limsup_{\nu \to 0} \nu \int |\nabla v_\nu|^2 > 0$.
• Physical Solutions: Solutions to the Euler equations that arise as weak limits of suitable Leray-Hopf solutions to the Navier-Stokes equations.
• The Gap in Literature: Previous constructions of anomalous dissipation (often using convex integration) typically resulted in dissipation measures that were singular in time (concentrated at a single time instant, e.g., $t=1$).
• The Open Problem: The authors aim to answer Questions 2.2 and 2.3 from [BDL23]: Can one construct a physical solution where the anomalous dissipation measure has a non-trivial absolutely continuous part with respect to the Lebesgue measure in time? This would align better with physical observations and simulations where dissipation occurs continuously over time.

2. Methodology

The authors employ a multi-step strategy combining advection-diffusion theory, quantitative homogenization, and convex integration-inspired constructions.

A. Reduction to Advection-Diffusion

The core of the proof relies on analyzing the advection-diffusion equation:
$$\partial_t \theta_\kappa + u \cdot \nabla \theta_\kappa = \kappa \Delta \theta_\kappa$$
where $u$ is a divergence-free autonomous velocity field in 3D ($T^3$). The authors construct a specific velocity field $u$ and initial datum $\theta_{in}$ such that the solution exhibits anomalous dissipation ($\limsup_{\kappa \to 0} \kappa \int |\nabla \theta_\kappa|^2 > 0$) with a time-continuous dissipation profile.

B. Construction of the Velocity Field

The velocity field $u$ is constructed based on the work of Crippa, Colombo, and Sorelli [CCS23], utilizing alternating shear flows with super-exponentially decreasing scales.

• Structure: The flow is designed to mix the scalar field $\theta$ efficiently in specific time intervals while maintaining high regularity ($C^\alpha$).
• Parameters: The construction involves a sequence of scales $a_q$ and time intervals $I_q$ defined by parameters $\alpha, \delta, \epsilon$. The flow is "autonomous" (time-independent in the 3D sense, though the construction involves time-dependent scaling in the proof logic).
• Key Property: The flow creates a "chessboard" pattern in the scalar field that is stretched and folded, leading to rapid decay of $L^2$ norms due to diffusion, even as $\kappa \to 0$.

C. Embedding into Navier-Stokes (The 4D Strategy)

To answer the questions for the 4D Navier-Stokes equations, the authors embed the 3D advection-diffusion solution into a 4D fluid system:

1. Dimension: They work on the 4-torus $T^4 \simeq T^3 \times T^1$.
2. Velocity Construction: They define a 4D velocity field $v_\nu = (u_\nu, \theta_\nu)$, where $u_\nu$ is a modified version of the 3D velocity field and $\theta_\nu$ is the scalar solution.
3. Forcing: They construct a body force $F_\nu$ that depends on $\nu$ but converges to a time-independent force $F_0$ in a specific Hölder norm ($C^\alpha$). This force is designed to make $v_\nu$ an exact solution to the forced Navier-Stokes equations.
4. Decoupling: The construction ensures that the dissipation of the 4D system is dominated by the dissipation of the scalar component $\theta_\nu$, effectively transferring the anomalous dissipation properties from the advection-diffusion equation to the Navier-Stokes equations.

D. Measure-Theoretic Analysis

The authors rigorously analyze the limit of the dissipation sequence $\nu |\nabla v_\nu|^2$. They decompose the limit measure $\mu$ into:

• An absolutely continuous part ($\mu_{ac}$) with respect to the Lebesgue measure in time.
• A singular part ($\mu_{sing}$).
  They prove that by carefully tuning the parameters, the singular part can be made arbitrarily small ($\|\mu_{sing}\|_{TV} < \beta$), while the absolutely continuous part remains non-trivial.

3. Key Contributions and Results

Theorem A: 4D Navier-Stokes and Euler

The main result establishes the existence of a weak physical solution $v_0$ to the 4D forced Euler equations and a sequence of smooth Navier-Stokes solutions $v_\nu$ such that:

1. Convergence: $v_\nu \rightharpoonup^* v_0$ in $L^\infty$.
2. Anomalous Dissipation: The sequence $\nu |\nabla v_\nu|^2$ converges weakly* to a measure $\mu$.
3. Time-Continuity: The projection of $\mu$ onto the time variable, $\mu_T = \pi_\# \mu$, has a non-trivial absolutely continuous part with respect to the Lebesgue measure.
4. Proximity to Duchon-Robert: The measure $\mu$ is close (in $H^{-1}_{t,x}$) to the Duchon-Robert distribution $D[v_0]$ of the limiting Euler solution.
5. Smooth Energy Profile: The kinetic energy of the limiting solution $e(t) = \frac{1}{2}\int |v_0|^2$ is smooth in time and strictly decreasing.

Theorem B: Advection-Diffusion Result

A foundational result proving the existence of an autonomous 3D velocity field $u \in C^\alpha$ and initial data $\theta_{in}$ such that the advection-diffusion solutions exhibit anomalous dissipation with a time-continuous measure. This serves as the engine for Theorem A.

Proposition 7.1 & Corollary 7.2: Distinction between Measures

The paper provides a counter-example showing that the Duchon-Robert distribution ($D[v_0]$) and the anomalous dissipation measure ($\mu$) are not always identical.

• They construct a case where $D[v_0] \neq 0$ (indicating energy loss in the limit) but the anomalous dissipation measure $\mu = 0$ (the vanishing viscosity limit does not capture the dissipation in the standard measure sense).
• This implies that for certain sequences, the vanishing viscosity limit is not a good approximation of the Euler solution in $L^p$ spaces, highlighting the subtlety of "physical solutions."

4. Significance

1. Resolution of Open Questions: The paper positively answers Questions 2.2 and 2.3 from [BDL23], proving that anomalous dissipation can be distributed continuously in time, not just concentrated at singular points.
2. Physical Relevance: The result aligns mathematical theory with physical experiments and simulations (e.g., [Sre84, PKvdW02]), which observe continuous energy dissipation in turbulent flows, rather than instantaneous bursts.
3. Refinement of Physical Solutions: By introducing the concept of "weak physical solutions" with time-independent forces (or forces converging uniformly), the authors provide a more robust framework for studying turbulence than previous convex integration examples which often relied on time-dependent, singular forces.
4. Dimensional Insight: The construction works in 4 dimensions. The authors note that while 3D examples exist, the 4D setting allows for the specific coupling of the velocity and scalar fields required to achieve the time-continuous dissipation property while maintaining the necessary regularity constraints.
5. Theoretical Nuance: The distinction drawn between the Duchon-Robert distribution and the anomalous dissipation measure deepens the understanding of the "Onsager conjecture" and the nature of singular limits in fluid dynamics. It suggests that the convergence of solutions does not automatically guarantee the convergence of their energy dissipation profiles in the expected manner.

Conclusion

Johansson and Sorella have successfully constructed a new class of solutions to the 4D Navier-Stokes equations that exhibit anomalous dissipation with a non-trivial absolutely continuous part in time. This bridges the gap between rigorous mathematical constructions (often singular) and the continuous nature of turbulence observed in physics, providing a significant step forward in the mathematical theory of the zeroth law of turbulence.