On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations

This paper demonstrates the failure of unconditional uniqueness for mild solutions to the Navier-Stokes and fractional Navier-Stokes equations in Besov spaces with negative regularity by constructing non-trivial stationary singular solutions via convex integration, while simultaneously establishing uniqueness for stationary weak solutions in an endpoint critical space.

The Gist

Imagine the Navier-Stokes equations as the ultimate "rulebook" for how fluids (like water, air, or even honey) move. For nearly a century, mathematicians have been trying to answer a simple question: If you know exactly how a fluid is moving right now, is there only one way it can move in the future?

Think of it like a movie. If you pause the film at a specific frame (the initial state), is there only one possible way the story can continue? Or could the movie branch off into two completely different, equally valid plotlines?

For a long time, mathematicians believed the answer was "Yes, there is only one story." This is called uniqueness. If the rules are clear, the future is determined.

However, this paper by Alexey Cheskidov and Hedong Hou says: "Not so fast." They have proven that for certain types of "rough" or "jagged" starting points, the fluid can actually split into two different futures. The story has two valid endings.

Here is how they did it, explained with some creative analogies.

1. The "Rough" Starting Point

In math, fluids are usually described as smooth, like a calm lake. But in the real world, fluids can be chaotic, turbulent, and full of tiny, violent swirls. The authors looked at fluids that are so "rough" they don't even have a defined energy level (mathematicians call this having "negative regularity").

Imagine trying to predict the weather. If you start with a perfectly smooth, calm atmosphere, the forecast is usually reliable. But if you start with a chaotic, jagged mess of wind and pressure, the rules of physics might allow for multiple different outcomes. The authors proved that in this "jagged" world, the fluid doesn't have to pick just one path.

2. The "Stationary" Ghost

To prove this, the authors didn't just watch a fluid move; they looked for a ghost.

They constructed a special kind of solution called a stationary singular solution.

• Stationary: It doesn't change over time. It's like a frozen wave.
• Singular: It's a "ghost" because it's so rough that it breaks the usual rules of energy. It's not a physical fluid you could hold in a bucket; it's a mathematical phantom.

The Analogy: Imagine a sculpture made of smoke. Usually, smoke dissipates. But imagine a "ghost sculpture" that stays perfectly still, frozen in time, yet is made of such chaotic smoke that it defies normal physics. The authors found that you can build many different versions of this ghost sculpture.

3. The Magic Trick: Convex Integration

How did they build these ghosts? They used a technique called Convex Integration.

The Analogy: Imagine you are trying to build a bridge across a canyon, but you are only allowed to use very specific, weird-shaped bricks.

1. You lay down a few bricks. The bridge is wobbly and doesn't quite reach the other side.
2. You add a new layer of bricks on top, but you wiggle them back and forth (oscillate) to fill in the gaps.
3. You repeat this process infinitely. With each layer, you make the bridge smoother and closer to the perfect shape, but you are constantly adding tiny, high-frequency wiggles.

In the end, you have a bridge that looks perfect from a distance, but up close, it's a chaotic mess of infinite wiggles. This process allows you to construct a solution that satisfies the equations but behaves in a wild, non-unique way. The authors used this "wiggle-building" technique to create their stationary ghosts.

4. The Two Movies

Here is the punchline of their discovery:

• Movie A: You start with a specific "rough" initial state (the ghost sculpture). The fluid just sits there, frozen in time, forever. This is a valid solution.
• Movie B: You start with the exact same initial state. But this time, the fluid wakes up! It starts moving, swirling, and evolving into a smooth, normal flow. This is also a valid solution.

Because both movies start with the exact same frame and both follow the laws of physics (the Navier-Stokes equations), the "unconditional uniqueness" has failed. The future is not determined; the fluid has a choice.

5. Why This Matters

This might sound like a weird mathematical curiosity, but it's huge.

• The "Critical" Line: For a long time, mathematicians thought that if you stayed within a certain "critical" range of roughness, the rules would hold firm. This paper shows that even in these critical zones, the rules break down.
• Fractional Equations: They also showed this happens even if you change the physics slightly (using "fractional" equations, which are like different versions of the fluid rules).
• The Limit: Interestingly, they also found a "safe zone." If the fluid is just smooth enough (in a specific mathematical space called $B^{-1}_{\infty,1}$), then uniqueness does return. It's like finding the exact point where the chaos turns back into order.

Summary

Think of the Navier-Stokes equations as a game of chess. For most of history, we thought that if you knew the position of all the pieces, there was only one legal move forward.

Cheskidov and Hou have shown that if the board is covered in a specific kind of "static" or "noise" (negative regularity), the game can branch into two different realities. One where the pieces stay still, and one where they start moving. Both are legal. Both are correct.

They didn't just find a glitch; they built a whole new world of "ghost fluids" to prove that the future of a fluid isn't always written in stone.

Technical Summary

Here is a detailed technical summary of the paper "On Non-Uniqueness of Mild Solutions and Stationary Singular Solutions to the Navier-Stokes Equations" by Alexey Cheskidov and Hedong Hou.

1. Problem Statement

The paper addresses the fundamental question of unconditional uniqueness for mild solutions to the Navier-Stokes equations (NSE) and the fractional Navier-Stokes equations (NSE$_\alpha$).

• Context: While local well-posedness is established in various critical and subcritical spaces (e.g., $L^d$, $\dot{B}^{-1+d/q}_{q,r}$, $BMO^{-1}$), the question of whether a solution is unique within the entire class of mild solutions (unconditional uniqueness) remains open or known to fail in specific supercritical regimes.
• Gap: Prior to this work, non-uniqueness results were largely restricted to supercritical spaces (e.g., $H^\beta$ for $\beta > 0$) or specific endpoint cases. The behavior of mild solutions in Besov spaces with negative regularity indices (including subcritical regimes) was not fully understood regarding uniqueness.
• Goal: The authors aim to prove that unconditional uniqueness fails for the NSE and NSE$_\alpha$ in all Besov spaces with negative regularity indices ($B^{-\theta}_{q,r}$ with $\theta > 0$), and to establish the existence of non-trivial stationary singular solutions that drive this non-uniqueness.

2. Methodology

The core methodology relies on Convex Integration, a technique pioneered by De Lellis and Székelyhidi and adapted for fluid dynamics by Buckmaster and Vicol. The authors construct non-trivial solutions through an iterative procedure.

A. Stationary Singular Solutions

Instead of constructing time-dependent non-uniqueness directly, the authors first construct non-trivial stationary singular solutions to the fractional Navier-Stokes equations:
$$\text{div}(u \otimes u) + (-\Delta)^\alpha u + \nabla p = 0, \quad \text{div } u = 0$$

• Definition of Singularity: A solution $u$ is "singular" if it is not in $L^2$ (infinite energy) and the tensor product $u \otimes u$ is not in $L^1$. Instead, $u \otimes u$ is defined as a paraproduct in the space of distributions modulo constants ($\dot{H}^{-s}$).
• Connection to Mild Solutions: They prove (Proposition 1.4) that any stationary singular solution $u$ generates a mild solution $U(t) = u$ to the time-dependent equation. Since the stationary solution is non-trivial and singular, it differs from the unique smooth solution generated by the same initial data (which exists in subcritical spaces), thereby proving non-uniqueness.

B. Iterative Convex Integration Scheme

The construction of the stationary singular solution involves an iterative scheme solving the Navier-Stokes-Reynolds system:
$$\text{div}(u \otimes u) + (-\Delta)^\alpha u + \nabla p = \text{div } R$$
where $R$ is the Reynolds stress tensor.

1. Perturbation: At each step $n$, given a solution $(u_{n-1}, R_{n-1})$, they construct a new solution $(u_n, R_n)$ by adding a velocity perturbation $w_n = u_n - u_{n-1}$.
2. Mikado Flows: The perturbation is built using Mikado flows (Daneri-Székelyhidi), which are highly oscillatory, divergence-free vector fields localized on specific geometric structures (lines).
3. Frequency Separation: The perturbation $w_n$ is constructed to have Fourier support in a dyadic shell far away from the support of $u_{n-1}$. This ensures that the nonlinear interaction terms (paraproducts) can be controlled and the Reynolds stress $R_n$ can be made arbitrarily small.
4. Convergence: By carefully choosing parameters (concentration $\mu$, oscillation $\gamma$, frequency $\lambda$), the sequence $u_n$ converges to a limit $u$ in various Besov and Lebesgue spaces, while the Reynolds stress $R_n$ vanishes.

C. Uniqueness in Endpoint Spaces

To complement the non-uniqueness results, the authors prove unconditional uniqueness for stationary solutions in a specific endpoint critical space ($B^{-1}_{\infty,1}$). This is achieved via energy flux estimates from high to low modes, showing that any solution in this regularity class must be smooth and thus zero (due to the stationary nature and mean-zero condition).

3. Key Contributions and Results

A. Failure of Unconditional Uniqueness (Theorems 1.1 & 1.2)

• Result: For the standard NSE ($\alpha=1$) and fractional NSE ($\alpha > 1/2$), unconditional uniqueness of mild solutions fails in all Besov spaces $B^{-\theta}_{q,r}$ with negative regularity index $\theta > 0$.
• Significance: This is the first non-uniqueness result in subcritical solution classes. Previously, non-uniqueness was only known in supercritical spaces. The result implies that even for initial data with very small norms in these spaces, multiple distinct mild solutions can exist.
• Scope: The result holds for any dimension $d \ge 2$ and any $\alpha > 1/2$ (for fractional NSE).

B. Existence of Stationary Singular Solutions (Theorem 1.5)

• Result: There exist infinitely many non-trivial stationary singular solutions to the fractional NSE.
• Regularity: These solutions belong to the intersection of all $L^p$ spaces for $1 \le p < 2$and all Besov spaces$B^{-\theta}_{q,r}$with$\theta > 0$.
• Extension: For $0 < \alpha < (d+1)/4$, these solutions also exist in$L^2(T^d)$, implying non-uniqueness of weak solutions in the energy space for these specific fractional powers.

C. Uniqueness in Critical Endpoint (Theorem 1.7)

• Result: There are no non-trivial stationary solutions in the critical endpoint space $B^{-1}_{\infty,1}$ (for $\alpha=1$) or its fractional analogues.
• Implication: This establishes a sharp boundary. While uniqueness fails in "worse" spaces (negative regularity), it is restored in this specific critical endpoint space, improving upon previous results by Lemarié-Rieusset.

D. Local Well-Posedness and Smoothing (Section 2)

• The authors establish local well-posedness for the fractional NSE in subcritical Lebesgue and Besov spaces, providing a rigorous framework for the "smooth" solution branch that coexists with the singular branch.

4. Significance and Impact

1. Resolution of Subcritical Uniqueness: The paper fundamentally changes the understanding of the Navier-Stokes equations by demonstrating that the "unconditional uniqueness" property, often assumed in subcritical regimes, is false. It shows that the solution map is not unique even when the initial data is small in norms that are traditionally considered "good" (subcritical).
2. Generalization of Convex Integration: The work extends the convex integration technique to stationary problems and fractional dissipation with arbitrarily large $\alpha$. This demonstrates the robustness of the method beyond the classical 3D NSE.
3. Sharpness of Critical Spaces: By proving non-uniqueness in negative Besov spaces and uniqueness in the specific endpoint $B^{-1}_{\infty,1}$, the paper delineates the precise regularity threshold where uniqueness breaks down.
4. Fractional Navier-Stokes: The results for $\alpha > 1$ are particularly novel, as the fractional Laplacian with high power usually implies stronger dissipation. The fact that non-uniqueness persists even for large $\alpha$ (in negative regularity spaces) highlights the dominance of the nonlinear transport term in low-regularity regimes.

Summary

Cheskidov and Hou utilize convex integration to construct stationary singular solutions with infinite energy but finite negative regularity. These solutions serve as counterexamples to unconditional uniqueness for the Navier-Stokes equations in all Besov spaces with negative regularity indices. This work bridges the gap between supercritical non-uniqueness and subcritical well-posedness, proving that the latter does not guarantee uniqueness, and establishes a sharp boundary for uniqueness in critical endpoint spaces.