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Density of weak solutions of the fractional Navier-Stokes equations in the smooth divergence-free vector fields

This paper establishes the density of weak solutions of the fractional Navier-Stokes equations in the space of smooth, divergence-free vector fields. By proving that any smooth, divergence-free field can be approximated arbitrarily closely by weak solutions with specific time-space regularity ( $L^s_t W^{1,q}_x$ ), the author utilizes convex integration to demonstrate non-uniqueness for a broad range of parameters.

Original authors: Michele Gorini

Published 2026-07-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Michele Gorini

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 a giant, invisible fluid swirling inside a box shaped like a donut (mathematicians call this a torus). This fluid is the subject of a famous puzzle: the Navier-Stokes equations. These equations are the rulebook for how fluids move, describing everything from smoke rising from a candle to blood flowing through veins. But here's the catch: while we know these rules exist, we aren't always sure if they lead to just one possible future or if they allow for multiple, equally valid futures starting from the exact same spot.

For a long time, mathematicians thought that if you started with a smooth, calm fluid, the rules would force it to evolve in only one way. However, mathematician Michele Gorini has proven that, under certain specific conditions, this isn't true. He showed that you can start with a perfectly smooth fluid and, using a clever mathematical trick, construct a scenario where the fluid suddenly splits into two completely different paths, both of which obey the rules perfectly. This work builds upon and generalizes earlier techniques developed by Cheskidov and Luo, extending the understanding of non-uniqueness that was first explored by researchers like Schnirelman, De Lellis, and Szekelyhidi.

The Magic Trick: Convex Integration

How did he do it? Think of it like a game of "whack-a-mole" with time.

Imagine you have a smooth, flowing river. The researcher wanted to introduce tiny, chaotic ripples that would eventually grow into a completely different river, but he had to do it without breaking the laws of physics. He used a technique called Convex Integration.

Picture this: You have a smooth sheet of clay (the fluid). You want to poke it with a needle to create a bump, but you can't just smash it; you have to fold it in a way that looks smooth from far away but is actually full of tiny, hidden wrinkles.

The Gluing: First, he took the smooth river and cut it into tiny time-slices. In the middle of each slice, he glued in a "corrector"—a tiny, fast-moving swirl that fixes the math for that specific moment.
The Sharp Cutoff: He used a "sharp cutoff" (like a super-fast switch) to turn these swirls on and off. Because the switch is so fast, the swirls only exist for a split second. To a slow observer, the fluid looks smooth. But to a fast observer, it's a chaotic mess of tiny, hidden movements.
The Fractional Twist: The paper deals with "fractional" Navier-Stokes equations. Imagine the fluid has a "friction" setting. In the standard version, the friction is set to 1. In this paper, he turned the friction dial to a different number (called $\theta$ ). He found that even with this different friction setting, you can still perform the magic trick of creating multiple futures.

The "Meta-Theorem" and the Rules

The author didn't just find one weird example; he proved a "Meta-Theorem." This is like a master key that says: "If you have a smooth starting fluid, you can always find a weird, non-unique solution, provided you follow a specific set of rules."

Crucially, the result shows that these weak solutions are dense in the space of smooth divergence-free vector fields. This means that any smooth, calm fluid flow can be approximated arbitrarily closely by these chaotic, non-unique weak solutions. It is not that smooth fields are dense among weak solutions, but rather that the chaotic weak solutions can mimic any smooth field as closely as desired.

These rules are a bit like a recipe for a cake that only works if you use specific amounts of flour and sugar. The paper lists these ingredients as mathematical parameters:

Time Integrability ( $s$ and $p$ ): These exponents describe how the solution behaves over time in Lebesgue-type norms. $s$ and $p$ measure the integrability of the solution in time.
Space Integrability ( $q$ ): This exponent describes how the solution behaves across space in terms of integrability.
Space Smoothness ( $\gamma$ ): This is the genuine measure of spatial smoothness, defined by the Sobolev exponent in the space $W^{\gamma,q}$ .

The paper proves that if you pick your numbers ( $s, p, q, \gamma, \theta$ ) within certain ranges, you can build these non-unique solutions. They even showed that these weird solutions are smooth (perfectly nice) for almost all of the time, except for a tiny, almost invisible set of moments where the chaos happens.

What They Ruled Out (and What They Didn't)

It is crucial to understand what this paper doesn't say.

It does NOT say the fluid is unpredictable in real life. The solutions he built are "weak solutions." Think of these as mathematical ghosts. They follow the rules of the equation, but they might not be the physical, "admissible" solutions that nature actually chooses. Nature might have a hidden rule that forces the fluid to pick just one path, even if the math allows two. The paper explicitly notes that for "admissible" solutions, the question of uniqueness is still a massive, unsolved mystery.
Admissibility means no energy gain. It is important to clarify that "admissible" does not mean the solutions obey strict energy conservation. Instead, it means they satisfy an energy inequality: the energy may decrease (due to friction), but it never increases. There is no energy gain. The paper does not claim that these non-unique weak solutions are admissible in this sense.
It does NOT say this happens for all friction settings. The paper focuses on specific ranges of the fractional exponent $\theta$ . It doesn't claim to have solved the problem for every possible value of friction.
It is NOT a simulation. This isn't a computer model that suggests non-uniqueness might happen. This is a rigorous mathematical proof. The author didn't just guess; he constructed the solutions step-by-step and proved they exist.

The Bottom Line

Michele Gorini has proven that for the fractional Navier-Stokes equations, the door to "multiple futures" is wide open for a specific class of mathematical solutions. He showed that if you are willing to accept solutions that are smooth most of the time but have tiny, hidden bursts of chaos, you can start with a calm fluid and end up with two different outcomes. Furthermore, he demonstrated that these chaotic weak solutions can approximate any smooth flow arbitrarily well.

However, the paper stops short of saying this means the real world is chaotic. It leaves the door open for the possibility that the "real" physical fluids (the admissible ones, which do not gain energy) still behave nicely and pick just one path. The mystery of whether the universe truly allows for multiple futures in fluid dynamics remains one of the biggest challenges in mathematics, but this paper has definitely added a new, fascinating chapter to the story by generalizing previous techniques.

Technical Summary: Density of Weak Solutions of the Fractional Navier-Stokes Equations in the Smooth Divergence-Free Vector Fields

Problem Statement
This paper addresses the problem of non-uniqueness for weak solutions to the fractional Navier-Stokes equations on the $d$ -dimensional torus $\mathbb{T}^d$ :
$\begin{cases} \partial_t v + \text{div}(v \otimes v) + \nabla p + (-\Delta)^\theta v = 0 \\ \text{div } v = 0 \end{cases}$
where $\theta \geq 0$ is the fractional dissipation exponent. While the classical case ( $\theta=1$ ) has seen significant progress regarding non-uniqueness via convex integration, the fractional case presents specific challenges. The presence of the fractional Laplacian term $(-\Delta)^\theta v$ complicates the control of the Reynolds stress, particularly when attempting to construct solutions with specific regularity properties in both space and time. It is important to note that the case $\theta=0$ corresponds to the fractional Navier-Stokes system with exponent 0, which retains the dissipation-like term and is distinct from the inviscid Euler equations, which lack such a term. Generally, the convex integration scheme becomes easier to implement as $\theta$ decreases.

The primary goal is to extend the non-uniqueness results of Cheskidov and Luo (specifically their density theorem for $L^1_t W^{1,q}_x$ solutions of the classical Navier-Stokes equations) to the fractional case. The paper aims to establish a "meta-theorem" demonstrating that smooth, divergence-free vector fields can be approximated arbitrarily closely by weak solutions with prescribed regularity classes $L^p_t L^\infty_x \cap L^s_t W^{\gamma, q}_x$ , where the parameters $p, s, q, \gamma$ are constrained by the fractional exponent $\theta$ and mutually limit each other within specific ranges.

Methodology
The proof relies on a convex integration scheme adapted from previous works on the Euler and Navier-Stokes equations (notably [2], [4], and [5]). The methodology is structured into an iterative scheme involving two main steps:

Concentration of Stress Error (Gluing):
Given a smooth solution $(u, R)$ to the fractional Navier-Stokes-Reynolds system, the time interval $[0, T]$ is subdivided into small intervals $I_i$ . On each interval, a corrector $v_i$ is constructed to solve the difference equation centered at $(u, R)$ , effectively making $u + v_i$ an exact solution on $I_i$ . A sharp cutoff function $\chi_i$ is applied to these correctors to glue them together. This process concentrates the stress error $R$ onto the boundaries of the sub-intervals, resulting in a new solution where the stress is supported on a set of small temporal measure. This step ensures the final solution has a singular set of small Hausdorff dimension.
Space-Time Convex Integration:
To reduce the size of the concentrated stress error, a perturbation $w$ is added to the glued solution. This perturbation is constructed using:
- Stationary Mikado Flows: Spatial building blocks ( $W_k$ ) supported on periodic cylinders, introduced in [7], which allow for $(d-1)$ -dimensional spatial concentration.
- Intermittent Temporal Functions: Time-dependent oscillations ( $g_\kappa, h_\kappa$ ) that modulate the spatial blocks.
The perturbation $w$ is decomposed into a principal part $w^{(p)}$ , a corrector $w^{(c)}$ to ensure incompressibility, and a temporal corrector $w^{(t)}$ . The design of these components balances the high-frequency spatial and temporal terms against the Reynolds stress. A key technical feature of this paper is the use of "convex integration in time," which allows for the balancing of high temporal frequency terms without requiring additional oscillation constraints, thereby enabling the construction of solutions with specific $L^s_t$ regularity.

Key Contributions and Results
The paper establishes Theorem 1.2.1, which provides a comprehensive set of conditions on the parameters $p, q, s, \gamma, \theta$ (and dimension $d$ ) under which the meta-theorem holds. Specifically, for any smooth, divergence-free starting field, there exists a weak solution $u$ that is $\epsilon$ -close to the starting field in the $L^p_t L^\infty_x \cap L^1_t W^{1,q}_x$ topology, while satisfying:

$u \in L^p_t L^\infty_x \cap L^s_t W^{\gamma, q}_x$ .
$u$ is smooth on a union of open intervals $I \subset [0, T]$ such that the complement (the singular set) has Hausdorff dimension $\leq \epsilon$ .
$u$ agrees with the starting field exactly at $t=0$ ; for small $t>0$ , $u$ agrees with the smooth solution arising from the initial datum of the smooth starting field.

Corollary 1.2.1 explicitly details the ranges of validity for these parameters. The results indicate a trade-off between time and space regularity. For instance:

If $\gamma < 1$ , the meta-theorem holds for various ranges of $q$ and $s$ depending on $p$ .
The paper recovers and extends previous results, such as the $L^1_t W^{1,q}_x$ non-uniqueness for $\theta=1$ (Cheskidov-Luo) and extends these to fractional $\theta$ and different time integrability classes $L^s_t$ .

The proof involves a delicate parameter choice (Section 6) to satisfy a system of inequalities derived from the scaling laws of the Mikado flows and the temporal oscillations. The author demonstrates that for appropriate choices of spatial frequency $\sigma$ , temporal frequency $\nu$ , concentration parameters $\mu, \kappa$ , and oscillation $\nu$ , the error terms can be made arbitrarily small.

Significance
The paper claims to extend the non-uniqueness landscape for the fractional Navier-Stokes equations. By generalizing the density result of Cheskidov and Luo from the classical case ( $\theta=1$ ) to the fractional case ( $\theta \in [1/2, 1]$ ) and broadening the time integrability from $L^1_t$ to $L^s_t$ , the work demonstrates that the obstruction to uniqueness is not solely due to the specific nature of the Laplacian but persists in the fractional regime.

The significance lies in the ability to construct weak solutions that are smooth almost everywhere (in the sense of Hausdorff dimension) yet fail to be unique, even when constrained to specific Sobolev regularity classes. This contributes to the understanding of the "Onsager conjecture" type phenomena in dissipative systems, showing that even with fractional dissipation, the energy inequality (admissibility) is not sufficient to guarantee uniqueness for solutions with low regularity. The paper does not claim to resolve the uniqueness of Leray-Hopf solutions (which satisfy the energy inequality) but rather focuses on the broader class of distributional weak solutions, highlighting the limitations of current regularity theories in ensuring uniqueness.

The Magic Trick: Convex Integration

The "Meta-Theorem" and the Rules

What They Ruled Out (and What They Didn't)

The Bottom Line

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