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# Navier-Stokes equations

## Latest Articles

### Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations

For more than 250 years, mathematicians have wondered if the Euler equations might sometimes fail to describe a fluid’s flow. A new computer-assisted proof marks a major breakthrough in that quest.

### Mathematicians Coax Fluid Equations Into Nonphysical Solutions

The famed Navier-Stokes equations can lead to cases where more than one result is possible, but only in an extremely narrow set of situations.

### Deep Learning Poised to ‘Blow Up’ Famed Fluid Equations

For centuries, mathematicians have tried to prove that Euler’s fluid equations can produce nonsensical answers. A new approach to machine learning has researchers betting that “blowup” is near.

### The Uselessness of Useful Knowledge

Today’s powerful but little-understood artificial intelligence breakthroughs echo past examples of unexpected scientific progress.

### Mathematicians Identify Threshold at Which Shapes Give Way

A new proof establishes the boundary at which a shape becomes so corrugated, it can be crushed.

### Latest Neural Nets Solve World’s Hardest Equations Faster Than Ever Before

Two new approaches allow deep neural networks to solve entire families of partial differential equations, making it easier to model complicated systems and to do so orders of magnitude faster.

### The Map of Mathematics

Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

### Mathematicians Prove Universal Law of Turbulence

By exploiting randomness, three mathematicians have proved an elegant law that underlies the chaotic motion of turbulent systems.

### For Fluid Equations, a Steady Flow of Progress

A startling experimental discovery about how fluids behave started a wave of important mathematical proofs.