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

In this 2014 blog post, renowned mathematician Terence Tao explores the mathematical behavior of the Navier-Stokes equations, which describe the motion of fluid substances. Specifically, Tao examines an averaged version of the three-dimensional Navier-Stokes equations to investigate the phenomenon of finite-time blowup—a scenario where a solution becomes infinite in a finite amount of time. The Navier-Stokes existence and smoothness problem remains one of the most significant unsolved challenges in physics and mathematics, carrying a Millennium Prize. Tao provides a detailed technical analysis, demonstrating that for the averaged system, solutions can indeed exhibit blowup. This work contributes to the broader theoretical understanding of fluid dynamics and the limitations of current mathematical models in predicting turbulent flow. By simplifying the equations through averaging, Tao offers insights into the potential mechanisms that might lead to singularities in the full, non-averaged three-dimensional Navier-Stokes equations.