Navier-Stokes equations

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

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.

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

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

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

Researchers have spent centuries looking for a scenario in which the Euler fluid equations fail. Now a mathematician has finally found one.

Turbulence is everywhere, yet it is one of the most difficult concepts for physicists to understand.

By squeezing fluids into flat sheets, researchers can get a handle on the strange ways that turbulence feeds energy into a system instead of eating it away.

The Navier-Stokes equations describe simple, everyday phenomena, like water flowing from a garden hose, yet they provide a million-dollar mathematical challenge.