One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Navier-Stokes Equations — Do Smooth Solutions for Fluid Flow Always Exist?
"We use these equations daily to design airplane wings — yet we don't even know if smooth solutions always exist." Equations from 1822, open for 200 years.
Connect a few background ideas before the real wall comes into view.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesClay Mathematics Institute — $1 million (Millennium Problem)
See the puzzle visually
Navier–Stokes existence and smoothness: do smooth initial conditions in a 3D fluid remain smooth forever, or can a finite-time singularity form? It is one of the seven Millennium Prize Problems.
Problem statement
Navier-Stokes: ∂u/∂t + (u·∇)u = -∇p + ν∇²u + f. Given smooth initial data in 3D, do smooth solutions exist for all time, or do they blow up in finite time?
The story of this puzzle
Navier (1822) and Stokes (1845) wrote the differential equations governing fluids. Every airplane and weather model uses them daily. Yet 200 years on, no one knows if 3D smooth solutions always exist or whether they blow up in finite time. The 2D case was settled by Leray in 1933. Tao showed in 2016 that modified NS equations can blow up — an ominous sign. Clay offers $1M. If blow-up exists, our entire approach to engineering is shaken; if not, one of math's deepest links to physics is sealed.
Try it yourself
Watch a bathtub vortex. The center has very high speed. Could speed go infinite in an infinitesimal region? That's the NS singularity question.
Beyond MathVoyage
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