Nobody knows whether the equations describing every breath of wind and splash of water can suddenly break down

The problem

The Navier–Stokes equations are the standard physics of flowing water and air — engineers trust them daily to design planes and forecast weather. Yet mathematicians cannot answer the most basic question about them: whether their 3D solutions always stay well-behaved, or whether a perfectly calm fluid can, in theory, concentrate energy so violently that the math breaks down at a single point in finite time. The two-dimensional version was tamed decades ago; the third dimension unleashes turbulence, whose cascade of swirls within swirls defeats every known technique. The question has been open since Leray's foundational work in 1934, and the Clay Institute made it one of its seven Millennium Prize Problems in 2000.

Why it matters

A regularity proof would certify that our fundamental model of fluids is mathematically sound, while a blowup counterexample would reveal that the equations predict physically impossible singularities and must be incomplete. Either outcome would transform the mathematics of turbulence — often called the greatest unsolved problem of classical physics — with consequences for aerodynamics, climate modeling, and beyond.

Progress so far

• 1934Jean Leray constructs global 'weak' solutions and proves smooth solutions exist for small initial data
• 1969Olga Ladyzhenskaya's work establishes that in 2 dimensions smooth solutions exist globally
• 2000Clay Mathematics Institute names it a Millennium Prize Problem
• 2016Terence Tao publishes finite-time blowup for an 'averaged' 3D Navier–Stokes equation, showing why many standard methods must fail on the real problem

References

1. Clay Mathematics Institute — official Millennium Prize problem page; status: unsolved
2. Quanta Magazine — coverage of Tao's averaged-equation blowup result and its implications