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Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014)

Terence Tao's 2014 paper "Finite time blowup for an averaged three-dimensional Navier-Stokes equation" demonstrates that certain averaged versions of the Navier-Stokes equations can exhibit finite-time blowup. This is significant because it invalidates "abstract" mathematical approaches to proving global regularity (the existence of solutions for all time) that rely solely on energy identities and general function space estimates for the nonlinear terms. Tao constructs an averaged nonlinearity that preserves the energy identity and function space bounds of the original Navier-Stokes equations but allows for solutions that grow infinitely large in finite time. This finding implies that any successful proof of global regularity for the true Navier-Stokes equations must exploit specific structural properties of the original nonlinearity, beyond what is captured by these general estimates. The work builds upon previous results for simplified or higher-dimensional models and suggests that finite-time blowup might be a possibility for the actual Navier-Stokes equations, at least for certain initial conditions.

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The core contribution of Tao's paper is the construction of a modified three-dimensional Navier-Stokes equation that, while preserving key properties like the energy identity and upper bound estimates on its nonlinear part, admits solutions that "blow up" in finite time. This is crucial because many theoretical approaches aiming to prove global regularity for the actual Navier-Stokes equations rely on these energy identities and general function space bounds. By showing a model equation that satisfies these conditions but still blows up, Tao effectively demonstrates that these general estimates are insufficient to guarantee global regularity and that any valid proof must leverage more specific features of the original Navier-Stokes nonlinearity.

This result carries significant implications for the mathematical community's pursuit of the Millennium Prize Problem concerning the existence and smoothness of solutions to the Navier-Stokes equations. It signifies a "supercriticality barrier," indicating that simplified, "abstract" analytical methods based on scaling arguments and energy bounds alone cannot resolve the problem. The paper suggests that proving global regularity would require exploiting deeper structural characteristics of the Navier-Stokes equations, such as their vorticity formulation, which are not captured by the averaged models. The construction itself involves a sophisticated averaging technique over spatial rotations and Fourier multipliers, creating a "dyadic model" analogous to those previously studied in higher dimensions.

The technical approach involves an "averaged" bilinear operator that mimics the behavior of the true Navier-Stokes nonlinearity in terms of function space estimates and energy conservation. Tao's construction is inspired by "local cascade operators" introduced by Katz and Pavlovic, which effectively reduce the problem to systems of ordinary differential equations in "wavelet coefficients." However, to overcome issues observed in previous dyadic models (where energy transfer cascades were interrupted, preventing blowup), Tao "engineers" an ODE system with a delayed but abrupt energy cascade mechanism. This engineered system demonstrates that finite-time blowup is possible under conditions that satisfy the energy identity and general estimates, thus blocking abstract proof strategies and hinting at the complexity of the true Navier-Stokes problem.

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Covered by: hackernews