🔢 analysis of PDEs

$L^2$ -Density of Wild Initial Data for the Hypodissipative Navier-Stokes Equations

This paper proves that for the three-dimensional hypodissipative Navier-Stokes equations with Laplacian exponent $\theta < 1/3$ , the set of $C^\beta$ Hölder continuous initial data (where $\theta < \beta < 1/3$ ) that generate non-unique dissipative weak solutions is dense in the space of $L^2$ divergence-free vector fields, thereby extending previous non-uniqueness results for the Euler equations and infinitely many wild initial data to a topologically stronger density statement.

Original authors: Michele Gorini

Published 2022-05-30

📖 6 min read🧠 Deep dive

Original authors: Michele Gorini

✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean of fluid, swirling with currents that we call wind or water. For over a century, mathematicians have tried to write a perfect rulebook for how this fluid moves. The most famous rulebook is called the Navier-Stokes equations. It's like the ultimate instruction manual for how a drop of water flows, how smoke curls, or how air rushes over a wing.

But here's the twist: nobody knows if this manual is unique. If you give the manual a specific starting picture (like a calm lake), does it produce exactly one future? Or could the same starting picture lead to two completely different futures, like a fork in the road where the fluid suddenly decides to swirl wildly in one version but stay calm in another?

This paper, written by Michele Gorini, dives into a slightly modified version of this rulebook. Instead of the standard friction that slows fluids down, the author introduces a "hypodissipative" version. Think of this as a fluid that is less sticky than normal, or perhaps one where the friction works in a weird, fractional way.

The Big Discovery: The "Wild" Start

The main finding of this paper is a bit like finding a "glitch" in the matrix of fluid physics. The author proves that for these less-sticky fluids, there is a huge, dense collection of starting conditions—called "wild initial data"—that lead to a total breakdown of predictability.

If you pick a starting state from this "wild" set, the math says you can't just have one future. You can have infinitely many different futures. It's as if you set up a domino chain, and depending on which invisible path the universe takes, the dominoes could fall in a straight line, spiral into a tornado, or dance a jig, and all of them would be mathematically valid according to the rules.

The paper shows that these "wild" starting points aren't rare, weird exceptions. They are everywhere. If you take any smooth, normal starting fluid state, you can wiggle it just a tiny bit (so small you can't even see the difference) and turn it into a "wild" state that explodes into infinite possibilities.

What the Paper Rules Out (The "No" List)

The paper is very careful about what it doesn't say. It does not prove that the standard, sticky Navier-Stokes equations (the ones we use for real-world weather and engineering) are unpredictable. It only proves this for the "hypodissipative" version, where the friction is weaker.

Furthermore, the paper argues against the idea that we can always guarantee a single, unique solution for all time. While we know that for a very short time, things might be predictable, the author shows that for these specific "wild" starts, the guarantee of a single path vanishes. The paper also clarifies that while we can prove these wild solutions exist, they are "weak" solutions. Think of them as solutions that work on a broad, statistical level but might have some rough edges or singularities, rather than being perfectly smooth and shiny everywhere.

How Sure Are We? (The Proof)

This isn't a guess, a simulation, or a computer model. The author has proved this mathematically. It's a rigorous, step-by-step logical construction.

To do this, the author uses a technique called Convex Integration. Imagine you are trying to build a sculpture out of clay, but you have a strict rule: the final shape must look like a specific smooth curve. Instead of molding it perfectly, you start with a rough block. Then, you add tiny, high-frequency ripples (like tiny waves) to the surface. These ripples are so fast and small that from a distance, the sculpture still looks smooth, but up close, they are doing all the heavy lifting to satisfy the complex rules.

The author builds a sequence of these "rough" solutions, adding more and more ripples, getting closer and closer to the final "wild" state. They prove that this process works and that the final result is a valid solution that behaves wildly.

The Trade-Off: Regularity vs. Global Existence

There is an important nuance regarding how long these solutions remain valid, and the paper is very honest about it. The author presents a mathematical trade-off rather than a hard time limit on the chaos itself.

On one hand, the author can construct solutions that maintain higher regularity (smoothness) for a fixed, finite amount of time. However, in this scenario, the guarantee that the solution adheres to the physical energy inequality (admissibility) holds only for a shorter, variable period.

On the other hand, the author can ensure that the solution is admissible (satisfies the energy inequality) for all time, extending to infinity. This is achieved by swapping the potentially inadmissible part of the solution with a standard Leray solution. The cost of this global admissibility is that the higher regularity is no longer guaranteed for the entire duration; it is only guaranteed on those smaller, initial time intervals.

So, the honest statement is a trade: you can have guaranteed higher regularity for a fixed finite time with admissibility only up to smaller, variable times, OR you can have admissibility guaranteed all the way to infinity, but with regularity only on those smaller times. The "wildness"—the property of the initial data admitting infinitely many solutions—remains a core feature of the starting condition, regardless of which side of this trade-off is chosen.

The Bottom Line

In simple terms, this paper says: "If you make the fluid friction weaker in a specific way, the universe of possible futures becomes incredibly messy. There are starting points everywhere that can lead to infinite different outcomes."

It doesn't break the real-world physics of water or air, but it breaks the mathematical hope that every version of fluid equations must have a single, unique answer. It shows that for a specific, slightly altered version of the rules, the future is not written in stone—it's written in ink that can be rewritten infinitely many ways, provided you start with the right kind of "wild" chaos.

Technical Summary: $L^2$ -Density of Wild Initial Data for the Hypodissipative Navier-Stokes Equations

Problem Statement
This paper addresses the Cauchy problem for the hypodissipative Navier-Stokes equations in the three-dimensional periodic setting ( $\mathbb{T}^3$ ). The system is given by:
$\begin{cases} \partial_t v + \text{div}(v \otimes v) + \nabla p + (-\Delta)^\theta v = 0 \\ \text{div} v = 0 \end{cases}$
where the dissipation exponent satisfies $\theta < 1/3$ . The author investigates the uniqueness of dissipative weak solutions (also called admissible or Leray solutions), which satisfy the energy inequality:
$\frac{1}{2} \int_{\mathbb{T}^3} |v(x,t)|^2 dx + \int_0^t \int_{\mathbb{T}^3} |(-\Delta)^{\theta/2} v(x,s)|^2 dx ds \leq \frac{1}{2} \int_{\mathbb{T}^3} |v(x,0)|^2 dx.$
While the existence of such solutions is known (via adaptation of Leray's classical arguments), the question of their uniqueness remains open. Previous works ([8, 20]) established the existence of infinitely many non-unique solutions for specific "wild" initial data in the Hölder class $C^\beta$ for $\theta < \beta < 1/3$ . However, the topological size of the set of such initial data was not fully characterized. This paper aims to prove that the set of initial data leading to non-uniqueness is dense in the space of divergence-free $L^2$ vector fields.

Methodology
The proof relies on the convex integration technique, a method pioneered by Nash and Kuiper for isometric embeddings and adapted to fluid dynamics by De Lellis and Székelyhidi. The strategy involves constructing a sequence of approximate solutions ("subsolutions") that converge to a weak solution.

Hierarchy of Subsolutions: The author defines three types of subsolutions:
- Strict subsolutions: Smooth solutions to the relaxed equation with a positive semi-definite Reynolds stress.
- Strong subsolutions: Subsolutions where the Reynolds stress is controlled by a power of its trace, allowing for iterative improvement.
- Adapted subsolutions: Strong subsolutions with specific blow-up rates of the $C^1$ norm as the Reynolds stress vanishes at $t=0$ , tailored to handle the initial data regularity.
Iterative Scheme: The core of the proof consists of two main approximation steps, repeated inductively:
- Gluing Step (Section 8): A localized procedure that patches together solutions on disjoint time intervals. This step utilizes a "gluing" technique (introduced in [25]) to ensure the Reynolds stress is supported on pairwise disjoint time intervals, preventing mixed terms in the perturbation. This step involves mollification and stability estimates for transport-diffusion equations.
- Perturbation Step (Section 9): A construction of high-frequency perturbations (Mikado flows) to absorb the Reynolds stress. Unlike the Euler case, the presence of the fractional dissipation term $(-\Delta)^\theta v$ requires careful control of the energy balance. The author introduces a "dissipative trace term" to manage the energy inequality, ensuring the constructed solutions remain admissible for a short time.
Parameter Control: The scheme involves a sequence of parameters $(\delta_q, \lambda_q)$ representing amplitude and frequency. The author introduces a new parameter $\Lambda$ and specific relations between exponents to control the interaction between the dissipation term and the convex integration errors. A critical technical challenge addressed is controlling the dissipation term in the energy inequality, which limits the time interval $T(\eta)$ over which admissibility can be guaranteed.

Key Contributions and Results

Theorem 1.1 (Existence of Wild Data): For any $\theta < \beta < 1/3$ and any $L^2$ initial datum $w$ , there exists a $C^\beta$ initial datum $w_\eta$ arbitrarily close to $w$ ( $\|w_\eta - w\|_{L^2} < \eta$ ) such that there exist infinitely many weak solutions $v_\eta$ starting from $w_\eta$ . These solutions satisfy the energy inequality on a time interval $[0, T(\eta)]$ , where $T(\eta) \to 0$ as $\eta \to 0$ .
Corollary 1.1 (Density for variable admissibility time): The set of $C^\beta$ initial data giving rise to non-unique admissible solutions is dense in the set of divergence-free $L^2$ vector fields.
Theorem 1.2 (Density of regular data producing irregular solutions): The set of initial data leading to non-unique solutions in the class $L^\infty_t H^\theta_x$ is also dense in the space of divergence-free $L^2$ vector fields.

Significance and Claims
The paper claims to improve upon previous non-uniqueness results for the hypodissipative Navier-Stokes equations ([8, 20]) by establishing the topological density of wild initial data, a property previously known only for the Euler equations ([14, 13]).

Generalization: The results generalize the density of wild initial data from the Euler equations to the hypodissipative Navier-Stokes equations.
Admissibility: The constructed solutions are dissipative (admissible), satisfying the energy inequality, which distinguishes them from non-admissible solutions where non-uniqueness is often easier to prove (e.g., [21]).
Relationship between Results: There is a distinct trade-off between regularity and admissibility time between the results. Corollary 1.1 provides higher regularity ( $C^\beta$ ) but only guarantees admissibility on a variable, shrinking time interval $T(\eta)$ . In contrast, Theorem 1.2 achieves global admissibility (for all time $t = \infty$ ) by swapping the potentially inadmissible part of the solution with a Leray solution. This modification makes the solutions global but comes at the cost of lower regularity ( $L^\infty_t H^\theta_x$ ).
Limitation on Time: The author explicitly notes that for the high-regularity solutions in Corollary 1.1, the admissibility condition (energy inequality) can only be guaranteed on a time interval $T(\eta)$ that shrinks to zero as the initial data approaches the target $L^2$ function. This is a necessary consequence of the need to control the dissipation term in the energy balance during the convex integration process.
Scope: The results hold for all $\theta < 1/3$ . The paper does not claim results for $\theta \geq 1/3$ , nor does it claim to resolve the uniqueness problem for all time or for all initial data, but rather demonstrates the prevalence of non-uniqueness within the class of admissible solutions for a dense set of initial conditions.

The work relies heavily on the constructive approach of convex integration, extending the "gluing" and "perturbation" strategies developed for the Euler equations to the setting of fractional dissipation, while navigating the additional constraints imposed by the energy inequality.

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