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Averaged Navier-Stokes equation blows up in finite time

Averaged Navier-Stokes equation blows up in finite time

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Tao's 2014 construction shows why the Millennium Prize problem resists standard PDE techniques

2 days ago: Hacker News discussion resurfaces Tao's paper

Overview

Updated 58 minutes ago

A $1 million prize problem in fluid dynamics has resisted proof for over two decades. Terence Tao's 2014 construction shows why: he built an averaged version of the Navier-Stokes equations that blows up in finite time, even though it conserves energy exactly like the real equations.

The result doesn't disprove global regularity for the actual equations, but it demonstrates the 'supercriticality barrier' that any proof must overcome. A 2026 preprint claims to have achieved finite-time blowup for the true equations, though the claim is unverified.

Why it matters

The Navier-Stokes global regularity problem carries a $1 million Millennium Prize; Tao's result shows why it resists standard PDE techniques.

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Key Indicators

2014
Year Tao published the averaged blowup result
Tao's paper appeared on his blog in February 2014 and in the Journal of the American Mathematical Society the same year.
$1M
Millennium Prize purse
The Clay Mathematics Institute offers $1 million for each of seven unsolved problems, including Navier-Stokes global regularity.
5/4
Critical α threshold for blowup in 2023 extension
The 2023 follow-up paper shows blowup for all α < 5/4 in the α-dissipative averaged equation.

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People Involved

Organizations Involved

Timeline

February 2014 September 2026

6 events Latest: 2 days ago
Tap a bar to jump to that date
  1. Hacker News discussion resurfaces Tao's paper

    Latest Discussion

    Tao's 2014 paper is discussed on Hacker News, likely in the context of the 2026 blowup claims.

  2. Preprint claims finite-time blowup for the true Navier-Stokes equations

    Claim

    A preprint (arXiv:2604.09949) claims to construct explicit initial data that blows up in finite time, settling the Millennium Prize problem in the negative. Unverified.

  3. Partial regularity paper extends Tao's result

    Research

    A follow-up paper proves blowup for α-dissipative averaged equations with α < 5/4, and bounds the blowup set's Hausdorff dimension.

  4. Regularization by noise paper studies Tao's model

    Research

    A paper analyzes whether stochastic transport noise can prevent blowup in Tao's averaged equation.

  5. Paper appears in the Journal of the American Mathematical Society

    Publication

    The paper is published in JAMS, one of the top mathematics journals.

  6. Tao publishes averaged Navier-Stokes blowup paper

    Publication

    Tao posts the paper on his blog, constructing an averaged Navier-Stokes equation that blows up in finite time.

Historical Context

3 moments from history that rhyme with this story — and how they unfolded.

2002-2003

Perelman's Poincaré conjecture proof (2003)

Grigori Perelman posted three papers proving the Poincaré conjecture, the first Millennium Prize problem to be solved. His proof was verified by multiple teams of mathematicians within a few years.

Then

Perelman was awarded the Millennium Prize in 2010 but declined it, along with the Fields Medal in 2006.

Now

It remains the only Millennium Prize problem solved, showing both the difficulty of these problems and the verification process for major claims.

Why this matters now

Perelman's proof shows what verified resolution of a Millennium Prize problem looks like, in contrast to the unverified 2026 Navier-Stokes claim.

2005

Katz-Pavlovic dyadic model (2005)

Nets Katz and Pavlovic constructed a dyadic shell model of the Navier-Stokes equations that blows up in finite time. The model simplifies the nonlinearity by averaging over frequency shells.

Then

The model provided evidence that the nonlinearity alone could drive blowup, absent the finer structure of the true equations.

Now

Tao's 2014 paper builds directly on this model, using a more complicated system of ODEs related to the Katz-Pavlovic construction.

Why this matters now

Tao's averaged equation is a sophisticated extension of the Katz-Pavlovic model, showing the same blowup mechanism survives in a setting that obeys the energy identity.

October 2013

Otelbaev's Navier-Stokes proof claim (2013)

Mukhtarbay Otelbaev, a Kazakh mathematician, posted a 100-page paper claiming to prove global regularity for the 3D Navier-Stokes equations. The claim drew intense scrutiny from the global mathematics community.

Then

Within months, mathematicians identified errors in the proof, including a flawed inequality in the final section.

Now

The episode reinforced skepticism toward unverified Navier-Stokes proofs and highlighted the difficulty of the problem.

Why this matters now

The 2026 preprint claiming finite-time blowup for the true Navier-Stokes equations faces the same verification gauntlet that Otelbaev's claim did.

Sources

(7)