Fluid dynamics

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.

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.

In the 1960s, drillers noticed that certain fluids would firm up if they flowed too fast. Researchers have finally explained why.

After decades of effort, mathematicians now have a complete understanding of the complicated equations that model the motion of free boundaries, like the one between ice and water.

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

Two teams found different ways for quantum computers to process nonlinear systems by first disguising them as linear ones.

Lauren Williams has charted an adventurous mathematical career out of the pieces of a fundamental object called the positive Grassmannian.

Having solved a central mystery about the “twirliness” of tornadoes and other types of vortices, William Irvine has set his sights on turbulence, the white whale of classical physics.

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