Terry Tao proves blowup for an averaged Navier-Stokes equation
A post on Terry Tao's blog describes a paper he uploaded to arXiv, "Finite time blowup for an averaged three-dimensional Navier-Stokes equation," submitted to the Journal of the American Mathematical Society. The paper's purpose is to formalize what the post calls the "supercriticality barrier": the claim that global regularity for the Navier-Stokes equation cannot be established by any abstract approach that relies only on upper-bound function space estimates for the equation's nonlinear term together with the energy identity. To make that precise, the paper builds an averaged version of the Navier-Stokes nonlinearity that satisfies essentially the same function-space estimates as the true equation and still obeys the energy identity, yet admits solutions that blow up in finite time. Since any proof strategy built purely from those estimates and the energy identity cannot tell the averaged equation apart from the real one, such a strategy cannot succeed against the real Navier-Stokes equation either.
Similar blowup results existed before this paper, but with gaps this one closes. Montgomery-Smith, Gallagher-Paicu, and Li-Sinai had shown blowup for Navier-Stokes variants that drop the energy identity. Katz-Pavlovic and Cheskidov had shown it for dyadic analogues that keep the energy identity, but only in five or more dimensions, with related work by Plechac and Sverak and by Hou and Lei pointing the same way. According to the post, this is the first blowup result for a Navier-Stokes-type equation in three dimensions that also satisfies the energy identity.
The construction is built from what the post calls "local cascade operators," combined into an ODE system for a sequence of wavelet-scale energy coefficients. Katz and Pavlovic's original dyadic model used a simpler version of this idea, but Barbato, Morandin, and Romito later showed that model actually has global smooth solutions in the dyadic case with non-negative initial data: energy cascading toward finer scales gets undercut because a scale starts leaking energy onward before it has absorbed all the energy meant for it. To get real blowup in three dimensions, the post describes engineering an ODE system, likened to a circuit built from "quadratic gates," that inserts a delay before each scale hands off its energy and then transfers it abruptly, so the cascade to finer and finer scales is not interrupted the way it was in the Katz-Pavlovic model. The post compares the resulting energy dynamics to the beacon-lighting scene in The Lord of the Rings, and to a self-replicating "von Neumann machine" that recreates itself at a finer scale and largely erases the previous copy, consistent with the energy identity.
Tao adds that the method of proof hints at a possible route to blowup for the true Navier-Stokes equations, and says he is now increasingly inclined to believe that is in fact the case, though only for a very small set of initial data. He calls it a real but remote possibility that the construction could be adapted to the true equations, speculating that if enough "logic gates" could be built out of an ideal, inviscid fluid, a von Neumann machine might in principle be built from the inviscid (Euler) form of the equations too, at which point the task would look more like a software engineering exercise than a PDE problem.
Key facts
- Terry Tao uploaded a paper, "Finite time blowup for an averaged three-dimensional Navier-Stokes equation," to arXiv and submitted it to the Journal of the American Mathematical Society
- The paper formalizes a "supercriticality barrier": no proof strategy that relies solely on upper-bound function space estimates for the nonlinear term plus the energy identity can establish global regularity for Navier-Stokes
- It is, per the post, the first blowup result for a three-dimensional Navier-Stokes-type equation that also satisfies the energy identity; earlier such results by Katz-Pavlovic and Cheskidov needed five or more dimensions
- The construction fixes a flaw Barbato, Morandin, and Romito found in the earlier Katz-Pavlovic dyadic model, by engineering a delayed, abrupt energy cascade between scales, described as a circuit of "quadratic gates"
- Tao says he is now more inclined to believe the true Navier-Stokes equations can blow up too, for a very small set of initial data, though he calls adapting the construction a real but remote possibility
Why it matters
The Navier-Stokes global regularity problem, whether smooth solutions to the real three-dimensional equations can always be continued forever, is one of the Clay Millennium Prize problems, and most attempts to solve it have leaned on upper-bound estimates for the nonlinear term combined with the energy identity. This paper shows that whole approach cannot work: it builds a modified equation that satisfies the same estimates and the same energy identity, yet still blows up in finite time, so no proof built only from those ingredients can distinguish the real equation from this one. It is also notable because Tao himself now says he leans toward believing the true equations can blow up, a stronger personal position than his earlier, more cautious writing on the topic.
Who it affects
This is a technical result for mathematicians working on partial differential equations and fluid dynamics, particularly anyone attacking the Navier-Stokes global regularity problem by the abstract, functional-analytic route the paper rules out. It has no direct bearing on engineering, weather modeling, or other applied uses of the Navier-Stokes equations, since the equation analyzed here is a deliberately altered, averaged version built to expose a limit on proof techniques, not the real fluid equation.
How to use it
The paper itself is on arXiv under the title given above and was submitted to the Journal of the American Mathematical Society; the blog post is a companion walkthrough of the ideas, including a circuit diagram of the "quadratic gates" and a sketch of how total energy behaves over time. There is no product, price, or license attached; the practical use is reading the construction to understand which proof strategies for Navier-Stokes are now known to be dead ends.
How solid is it
The construction directly extends and repairs earlier work: Katz and Pavlovic's dyadic cascade model, refined by Cheskidov, already gave blowup in five or more dimensions, and Barbato, Morandin, and Romito had identified exactly why that dyadic model fails to blow up in three dimensions with non-negative data. The paper's fix, a delayed and abrupt cascade between scales, is presented as closing that specific gap. That said, the post omits the full formal statement of the paper's main theorem, including the precise integrability conditions required on the Fourier multipliers used, so parts of the argument cannot be checked from the post alone, and the text does not say whether the Journal of the American Mathematical Society submission was subsequently accepted.
Risks and caveats
Tao is explicit that his belief that the true Navier-Stokes equations can blow up is a personal inclination, not something this paper proves, and he restricts even that belief to a very small set of initial data. He similarly describes any path from this averaged construction to the real equations as a real but remote possibility, dependent on speculative ideas like building a von Neumann machine out of an ideal, inviscid fluid. The averaged equation analyzed in the paper is a deliberately modified nonlinearity built to expose a barrier in proof techniques; it is not the true Navier-Stokes equation, and this result does not resolve the Millennium Prize problem.
“There is a real (but remote) possibility that this sort of construction can be adapted to the true Navier-Stokes equations.”
— Terry Tao