Can fluctuations break it?

Four ways to make the equations fluctuate, and what each does to the constructed singularity.

The construction is a statement about one equation with one force. A real fluid is not that equation. It has thermal noise, a molecular scale, and unresolved fast motions. This page asks whether a model with any of those, arguably the more accurate model, would still blow up under the constructed force. The answer depends on what "fluctuating" means, and in one case it is a theorem.

Three senses of breaking

A proof can be broken by finding an error. This page does not do that and has no reason to think one exists; the Lean formalization makes a gap in the logic unlikely, though the fidelity of the formal statement to Fefferman's is a separate human check. A proof can also be shown to rest on hypotheses that the physical system does not satisfy. That is easy here, and it is not news: the force is smooth but not analytic, the fluid is at zero temperature, and the continuum has infinite resolution. The interesting sense is the third. Take a model that differs from Navier–Stokes by a fluctuation, and ask whether the constructed force still produces a singularity. If not, the theorem is true and irrelevant to that model, and the question becomes which model is the fluid.

What the construction depends on

Three features of the proof decide how it responds to a perturbation.

• The force is the residual. It is defined as what the equation leaves over after the flow is chosen. Change the equation and the same force no longer matches the same flow; the mismatch is the size of the change times the size of the flow's derivatives, and those diverge. A perturbation of relative size $\epsilon$ in the viscous term produces a residual $\epsilon\,\nu\Delta u\asymp\epsilon\,\tau^{-3/2-h}$, the same order as the terms the construction spent 166 pages cancelling.
• The seeds are exponentially small. Each pulse starts at a Gaussian tail, $P(0)\le e^{-cL_s/4}$ with $L_s\asymp(\log 1/q)^2$, and is amplified by the shear to an amplitude comparable to the background. Anything else that seeds the same instability at larger amplitude gets the same amplification.
• The cancellation is to every order. The residual is made flat, smaller than every power of $q$. A perturbation that is any fixed power of $q$ is not small on that scale.

In English: the singularity is an exactly balanced object, not a stable attractor of the dynamics. That does not make it wrong. It makes it fragile, and the question is whether fragility can be turned into a theorem for a given fluctuating model.

Model 1: a fast exogenous switch

Suppose the viscosity flips between two values at rate $\lambda$, driven by a Markov chain that knows nothing about the flow. This is the fast switching of the homogenization pages. For $\lambda$ large the averaged equation is Navier–Stokes with the mean viscosity plus a corrector of order $1/\lambda$, and the theorem holds at the mean viscosity with its own force. Nothing is broken.

Except that averaging requires the switch to be fast relative to the dynamics, and a singularity has no slowest time scale. The core evolves on the time remaining, $\tau$. Once $\tau\ll1/\lambda$ the switch is slow: the flow sees a single frozen value of $\nu$ for the rest of its life. The force was tuned to the mean viscosity and now the equation has a different one, and by the first point above the pre-tuned force is wrong by a term of the leading order. The singularity outruns the averaging. Whether the switched equation admits any fixed force that blows up along almost every path of the chain is open. Cao, Chi and Nie show that from any smooth state there are nearby forces that produce blow-up, but a fixed force must be chosen before the path is known, so their density result does not answer it.

Model 2: transport noise

Replace the unresolved small-scale velocity by a stochastic transport term: fluid parcels move with the resolved velocity plus a rapidly fluctuating, divergence-free, Stratonovich noise with a prescribed spectrum. This is the model of Flandoli and Luo, and of Agresti, and it is derived from Newton's law with a separation of scales. Here there is a theorem. Agresti proves that for any bound $M$ on the data and any $\epsilon>0$ there is a noise intensity and spectrum for which the equation on the torus has a unique global smooth solution with probability at least $1-\epsilon$, and states in his Remark 4.4 that the result extends to a deterministic force $f$ that is smooth with spatial derivatives bounded uniformly in time.

The constructed force qualifies: it is $C_c^\infty$. So the periodic equation with the constructed force and a suitable transport noise has a global smooth solution with high probability. For that equation the conclusion of the theorem fails. The mechanism is the one the first point predicts: in the scaling limit the transport noise acts as an additional eddy viscosity with enhanced dissipation, the flow under the fixed force is a different flow, and the fine balance is gone. There is no contradiction with the theorem holding for every $\nu$, because the theorem's force depends on $\nu$ and the eddy viscosity was not in the force's design.

The caveats are real. The noise is chosen after the force, with intensity depending on its size; whether a fixed physical noise defeats every force is a different and open question, and the natural next construction would be a force built to blow up through the noise. The noise is multiplicative and scale-invariant, a model of turbulent transport, not of molecular motion. And "with high probability" is not "surely". Still, this is the one fluctuating model in which the constructed singularity is known not to survive.

Model 3: thermal noise

Landau and Lifshitz add to the viscous stress a random stress, white in space and time, with variance $2k_BT\,\rho\nu$ per unit volume and time, fixed by the fluctuation-dissipation relation. This is fluctuating hydrodynamics and it is the physically correct correction to Navier–Stokes at scales above the mean free path. It is also, as a PDE in three dimensions, too rough for classical solutions, and results on regularization by additive noise are described by Flandoli and Luo as inconclusive. There is no theorem either way for the constructed force.

There is a heuristic. The velocity fluctuation thermal noise induces at scale $\ell$ is $v_T(\ell)=\sqrt{k_BT/\rho\ell^3}$, and relative to the core speed $\nu/\ell$ this is $\sqrt{k_BT/\rho\ell}\,/\nu$, a fraction that grows as the core shrinks and reaches about a percent at the mean free path in air. That is many orders of magnitude above the exponentially small seeds of the construction, at every scale the construction uses. The seeds page puts the two numbers side by side. Thermal noise projected onto the growing direction of the pulse equation is amplified by the same factor the designed seed is, and if it reaches the background amplitude the linearization fails and the annulus is turbulent rather than tuned. This is not a proof that the thermal equation does not blow up; it is a reason to expect that the constructed solution is not what the thermal equation does under the constructed force.

Model 4: molecules

Below the mean free path there is no fluid. Navier–Stokes is the first-order Chapman–Enskog approximation to Boltzmann in the Knudsen number $\ell_{\rm mfp}/\ell$, and it is not the more accurate model at small $\ell$; Boltzmann is. The core radius is $\sqrt{\nu\tau}$ up to a fixed factor and reaches the mean free path at $\tau^*=\ell_{\rm mfp}^2/\nu$, a third of a nanosecond before the singular time in air. At that moment the core speed $\nu/\ell_{\rm mfp}$ is the molecular thermal speed, so the Mach number is order one as well; this is not a coincidence but the kinetic-theory identity $\nu\approx\tfrac12\ell_{\rm mfp}\bar v$. The continuum, the incompressibility, and the isothermal assumption all fail together. Braunstein's point that a molecular spectral cutoff turns the thermal equation into a finite-dimensional stochastic system with a Lyapunov function, hence globally well posed, is the same observation from the other side. The molecular limit page has the arithmetic for air and water.

The averaging trap

One tempting shortcut fails and is worth displaying. Suppose one replaced the velocity by its angular mean and asked whether the averaged equation blows up. The construction's own stress is the answer: the pulses average to zero and their momentum flux $\langle w\otimes w\rangle$ does not, and that flux is a leading-order term. Replacing $u$ by $\bar u$ drops it. The proof is a homogenization calculation run backwards, choosing the fast oscillations to produce a wanted Reynolds stress, and any averaged model that omits the stress is not the averaged equation. The $1/\lambda$ Green–Kubo corrector of an exogenous chain does not apply to oscillations the flow generates itself, without a separate scaling theorem.

Verdict

Neither reading dominates. The theorem is correct about its equation and the equation is not the fluid at the scale where the theorem bites. The fluctuating models that are known to be regular achieve it by tuning the noise to the force, which is the same move the theorem makes in the other direction. The question that would settle more is whether a fixed physical noise defeats every smooth force, and nobody has proved that.

What would settle more

• A transport-noise theorem with the noise fixed first and the force arbitrary, or a construction of a force that blows up through a fixed noise. Either would decide model 2.
• A numerical integration of the linearized pulse equation of the pulses page with thermally seeded initial data, to see whether the seeded modes saturate before the designed ones deliver their stress. The seeds page is the back of the envelope for that.
• The constants. The theorem never states the prefactor $e_0$ or the profile scales, and the molecular estimates on the molecular limit page set them to one. A construction with enormous constants would move the continuum failure earlier, not later.

Next: the molecular limit.