Navier–Stokes existence and smoothness

What makes this fascinating

• The math of every fluid — These equations describe water, weather, and blood flow — engineers rely on them daily.

• Do solutions blow up? — No one can prove that smooth starting conditions never produce infinite velocities — a singularity — in finite time.

• A Millennium Prize Problem — Proving solutions always stay smooth (or finding one that doesn't) is worth $1,000,000.

Frequently asked questions

What is the Navier–Stokes existence and smoothness problem?

It asks whether the Navier–Stokes equations, which model fluid flow, always have smooth, well-defined solutions in three dimensions — or whether they can 'blow up' to infinite values in finite time. It is one of the seven Clay Millennium Prize Problems.

Has it been solved?

No. We use the equations successfully every day in engineering, but a mathematical proof that their solutions always stay smooth — or a counterexample — has never been found.

Why does it matter?

A blow-up would mean the equations break down and miss some physics; a proof of smoothness would put the mathematics of turbulence on rigorous footing. It carries a $1,000,000 prize.

More summits in Mathematics

• Riemann Hypothesis

  A 160-year-old pattern in the primes that no one can prove — math's most famous open problem.

• P vs NP

  If a solution is easy to check, is it always easy to find? A million-dollar question at the heart of computing.

• Birch and Swinnerton-Dyer conjecture

  A startling claim that a curve's whole-number solutions are encoded in a single function.

• The Collatz conjecture

  A rule a child can follow, an answer no mathematician can prove. Start anywhere — do you always reach 1?