GoKawiilTerence Tao constructs a Navier-Stokes variant that blows up in finite time
Terence Tao posted a paper showing that a modified, 'averaged' version of the three-dimensional Navier-Stokes equations can develop a finite-time singularity, even though this modified equation preserves the same energy identity and function-space bounds as the real equations. Earlier blowup examples for Navier-Stokes-like systems either lacked the energy identity or required five or more dimensions; Tao's construction is the first to combine three dimensions, the energy identity, and finite-time blowup.
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The result formalizes a 'supercriticality barrier': it proves that no purely abstract method relying only on energy estimates and upper bounds on the nonlinearity can ever prove global regularity for Navier-Stokes, since such methods would equally apply to this blowing-up variant. Tao notes the proof technique hints at a possible strategy for showing the actual Navier-Stokes equations could blow up for some very special initial data, a step toward resolving one of the Clay Millennium Problems.
- The averaged model shares energy identity and nonlinearity bounds with true Navier-Stokes, yet still blows up.
- This rules out any proof strategy for global regularity based solely on such abstract estimates.
- The construction suggests a possible pathway toward proving blowup in the genuine 3D Navier-Stokes equations.
Navier-Stokes Equations by Roger Temam — If Tao's blowup construction for averaged Navier-Stokes has you curious, Temam's classic text is the standard entry point into the theory and numerical analysis of these equations. It's a rigorous reference that covers the functional-analytic setting behind energy identities and regularity questions.
See Navier-Stokes Equations by Roger Temam on Amazon → Affiliate link — we may earn a commission on purchases, at no extra cost to you. Product picked by AI based on this article; it is not a tested recommendation.Source: terrytao.wordpress.com — Terence Tao, 2026-09-05
Published there as: “Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014)”
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