Partial differential equations

By mathematically proving how individual molecules create the complex motion of fluids, three mathematicians have illuminated why time can’t flow in reverse.

Turbulence is a notoriously difficult phenomenon to study. Mathematicians are now starting to untangle it at its smallest scales.

A powerful mathematical technique is used to model melting ice and other phenomena. But it has long been imperiled by certain “nightmare scenarios.” A new proof has removed that obstacle.

On its surface, the Kakeya conjecture is a simple statement about rotating needles. But it underlies a wealth of mathematics.

A new proof marks major progress toward solving the Kakeya conjecture, a deceptively simple question that underpins a tower of conjectures.

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.

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 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.