Non-uniqueness of smooth solutions of the Navier-Stokes equations from almost the same initial conditions

This paper presents numerical evidence using Clean Numerical Simulation suggesting that the Navier-Stokes equations may admit distinct global solutions arising from initial conditions differing by as little as $10^{-40}$, thereby challenging the uniqueness of smooth solutions and offering new insights into the associated Millennium Prize Problem.

The Gist

The Big Question: Can Two Almost-Identical Starts Lead to Totally Different Futures?

Imagine you are baking two cakes. You use the exact same recipe, the same oven, and the same ingredients. But, in one cake, you add a single grain of salt that is so tiny it's invisible to the naked eye—let's say, a grain that is one ten-billionth of a gram smaller than the other.

In the real world, those two cakes would taste exactly the same. But in the chaotic world of fluid dynamics (how water and air move), this paper suggests something mind-blowing: Those two cakes could end up looking and tasting completely different.

The Problem: The "Butterfly Effect" and Computer Errors

For decades, scientists have struggled to prove that the Navier-Stokes equations (the math rules that describe how fluids like water and air flow) always have one unique solution for a given starting point. This is a huge mystery, so big that solving it is worth a $1 million prize (the Millennium Prize).

The problem is that fluids are chaotic. This is the famous "Butterfly Effect": a tiny flap of a butterfly's wings in Brazil can theoretically cause a tornado in Texas.

When scientists try to simulate this on computers, they run into a wall. Computers aren't perfect; they make tiny rounding errors (like writing 1/3 as 0.333333). In a chaotic system, these tiny computer errors act like that butterfly's wings. They grow exponentially fast, turning into massive errors that make the simulation useless. It's like trying to predict the weather a month from now; the tiny errors in your data make the prediction wrong almost immediately.

The Solution: "Clean Numerical Simulation" (CNS)

The authors, Shijun Liao and Shijie Qin, developed a super-advanced way of doing math called Clean Numerical Simulation (CNS).

Think of a normal computer simulation as a standard flashlight. It's good enough to see the room, but if you look closely, you see the grainy noise in the light.
Their CNS is like a laser beam in a vacuum. They used special math tricks and super-precise numbers (with hundreds of digits of accuracy) to make the "noise" (computer errors) so small that they are essentially zero for a long period of time.

They managed to keep the simulation "clean" for a long time, long enough to see what the fluid actually does, rather than what the computer thinks it does.

The Experiment: The Tiny Push

They set up a simulation of a swirling fluid (a "Kolmogorov flow") in a square box.

1. Scenario A: They started with a specific, perfectly symmetrical swirl.
2. Scenario B: They started with the exact same swirl, but added a tiny, invisible nudge (a mathematical term so small it's like $10^{-40}$). To put that in perspective: if the fluid were the size of the entire universe, this nudge would be smaller than a single atom.

The Result:

• At the start: Both fluids looked identical.
• After a while: The tiny nudge in Scenario B grew. Because the fluid is chaotic, that microscopic nudge exploded into a massive change.
• The End: The two fluids ended up in completely different states.
  • Scenario A kept its perfect symmetry (like a spinning top).
  • Scenario B lost its symmetry and became a messy, different pattern.
  • Even the "statistics" (the average energy and behavior) of the two fluids were totally different.

The Analogy: The Tightrope Walker

Imagine a tightrope walker balancing perfectly on a wire.

• Fluid A is the walker standing perfectly still.
• Fluid B is the same walker, but someone blows a breath of air so faint it's undetectable.

In a normal world, the walker wouldn't notice. But in this chaotic fluid world, that faint breath causes the walker to stumble, spin, and fall in a completely different direction than if they had just stood still.

Why Does This Matter?

This paper provides numerical evidence that the Navier-Stokes equations might not have a unique solution.

Usually, we believe that if you know the starting point of a system perfectly, you can predict exactly what happens next. This paper suggests that even if you know the starting point almost perfectly (differing by a microscopic amount), the future can be completely different.

• The "Clean" Part: Because they used such precise math, they are confident this isn't just a computer glitch. It's a real property of the equations.
• The "Non-Uniqueness" Part: They showed that two starting points that are practically identical can lead to two totally different global outcomes.

The Bottom Line

The authors aren't saying they have solved the $1 million math problem yet. Instead, they are saying: "Hey mathematicians, look at this! We built a super-precise microscope, and it shows that these fluid equations might allow for two different futures from the same starting line. Here is the data; maybe this gives you a clue on how to prove it."

It's a roadmap for proving that in the chaotic world of fluids, the future might be far less predictable than we thought.

Technical Summary

1. Problem Statement

The paper addresses a fundamental open problem in mathematics and fluid dynamics: the uniqueness and existence of smooth solutions to the Navier-Stokes (NS) equations, which is one of the seven Millennium Prize Problems.

• The Core Issue: While it is widely accepted that turbulent flows are governed by NS equations, it remains unproven whether a unique smooth solution exists for all time ($t > 0$) given specific initial conditions.
• The Challenge: Previous theoretical work (e.g., Coiculescu and Palasek) has shown non-uniqueness in critical spaces with infinite energy, but explicit numerical examples of non-uniqueness for finite energy solutions have been elusive.
• The Obstacle: Traditional numerical methods, such as Direct Numerical Simulation (DNS), suffer from the "butterfly effect." In chaotic systems like turbulence, minute numerical errors (round-off and truncation) grow exponentially, causing the simulation to diverge rapidly from the exact trajectory. This makes it impossible to distinguish between physical sensitivity to initial conditions and artificial numerical noise.

2. Methodology: Clean Numerical Simulation (CNS)

To overcome the limitations of traditional DNS, the authors employ Clean Numerical Simulation (CNS), a high-precision algorithm developed by Liao.

• Mechanism: CNS reduces both truncation errors and round-off errors to negligible levels by using:
  • Multiple-precision arithmetic: Variables and parameters are calculated with a high number of significant digits ($N_s$).
  • High-order Taylor expansion: Temporal integration uses high-order expansions ($M$).
  • Fourier pseudo-spectral method: Used for spatial approximation on a uniform mesh.
• Validation Strategy: To ensure reliability, the authors calculate a "critical predictable time" ($T_c$). They run a second, even more precise CNS simulation (with higher $M$ and $N_s$) to verify that the primary simulation's numerical noise is rigorously negligible throughout the interval $t \in [0, 300]$.
• Governing Equations: The study focuses on a 2D incompressible Kolmogorov flow in a square domain $[0, 2\pi] \times [0, 2\pi]$ with periodic boundary conditions.
  • Reynolds number ($Re$): 2000
  • Kolmogorov forcing scale ($n_K$): 16
  • Equation: Dimensionless stream function form of the NS equations.

3. Experimental Setup

The authors designed a comparative experiment using four distinct initial conditions that are "almost the same" but possess different spatial symmetries:

1. Flow CNS (Reference): Initial condition $\psi_1$ containing both rotation and translation symmetries.
2. Flow CNS'$_1$, CNS'$_2$, CNS'$_3$: Initial conditions $\psi_2$ defined as $\psi_1 + \delta \sin(x+y)$.
   • The perturbation $\delta$ is set to extremely small values: $10^{-10}$, $10^{-20}$, and $10^{-40}$.
   • These perturbations break the rotation symmetry, leaving only translation symmetry.
   • The magnitude of the difference between initial conditions is $|\psi_2 - \psi_1| \leq |\delta|$.

4. Key Results

Using CNS, the authors demonstrated that solutions starting from these nearly identical initial conditions diverge into distinct global solutions:

• Divergence of Trajectories:
  • Initially, the perturbed flows (CNS'$_{1,2,3}$) track the reference flow (CNS) closely.
  • After a certain duration (around $t > 100$), they diverge significantly.
  • Metric: The spatially averaged kinetic energy dissipation rate $\langle D \rangle_A$ of the reference flow becomes nearly twice as large as that of the perturbed flows.
• Symmetry Breaking:
  • The reference flow (CNS) retains the rotation and translation symmetry of its initial condition.
  • The perturbed flows (CNS'$_{1,2,3}$) lose the rotation symmetry and retain only translation symmetry.
  • This confirms that the tiny initial disturbance ($\delta$) was amplified by the chaotic nature of turbulence ("noise-expansion cascade") to a macroscopic level, fundamentally altering the flow's global properties.
• Statistical Differences:
  • Probability Density Functions (PDFs) of the kinetic energy dissipation rate differ significantly between the reference and perturbed flows.
  • Despite the initial conditions being identical up to 40 decimal places (in the case of $\delta = 10^{-40}$), the resulting statistical states are distinct.
• Finite Energy: All solutions possess finite energy density, distinguishing this work from previous non-uniqueness proofs involving infinite energy.

5. Significance and Contributions

• Numerical Evidence for Non-Uniqueness: The paper provides explicit numerical evidence that the Navier-Stokes equations can admit distinct smooth global solutions from initial conditions that differ by an arbitrarily small amount ($\delta \to 0$).
• Overcoming Numerical Noise: It demonstrates that with CNS, one can simulate chaotic turbulence long enough to observe physical divergence rather than numerical artifact, effectively bypassing the limitations of standard DNS.
• Implications for the Millennium Prize Problem: The results challenge the assumption of uniqueness for smooth solutions with finite energy. The authors propose a conjecture: The Navier-Stokes equations admit distinct smooth global solutions for all $t > 0$ from almost the same initial conditions $U_1$ and $U_2$ where $|U_1 - U_2| \leq \delta$ for arbitrarily small $\delta$.
• Roadmap for Mathematicians: These examples serve as a "roadmap" or counter-examples for mathematicians attempting to prove or disprove the uniqueness of NS solutions, suggesting that uniqueness may not hold even for finite-energy, smooth initial data.

Conclusion

The study utilizes high-precision Clean Numerical Simulation to show that in 2D turbulent Kolmogorov flow, infinitesimal changes in initial conditions (as small as $10^{-40}$) lead to macroscopically distinct global solutions with different symmetries and statistical properties. This provides strong numerical support for the non-uniqueness of smooth Navier-Stokes solutions, offering a critical perspective on one of the most difficult problems in mathematical physics.