A New Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations

Calculating and modeling the motion of fluids has been done by mathematicians for hundreds of years. Researchers have been aided by the equations that describe how ripples crease the surface of a pond. When written in the right language, these equations are easy to understand. Making sense of basic questions about their solutions can be hard.

The oldest and most prominent of these equations was formulated more than 250 years ago. According to a mathematician at Duke University, almost all of the fluid equations are derived from the equations in the book. You could say they're the first ones.

It is not known whether the equations are an accurate model of ideal fluid flow. One of the biggest problems in fluid dynamics is figuring out if the equations ever fail, outputting nonsensical values that make it impossible to predict a fluid's future states.

There are initial conditions that cause the equations to break down, according to mathematicians. They haven't been able to show it.

A pair of mathematicians have shown that a particular version of the equations sometimes fails. While the proof doesn't completely solve the problem for the general version of the equations, it does offer hope that a solution is within reach. The mathematician who was not involved in the work said it was an amazing result. In the literature, there are no results of its kind.

There's only one thing left to do.

Significant use of computers is made by the 177-page proof. It's hard for other mathematicians to verify it. Many experts think the new work will be correct, but they are still in the process of doing so. They have to think about what a proof is and what it will mean if computers are the only way to solve important questions in the future.

Looking at the beast.

If you know the location of the particles in the fluid, the equations should be able to predict how the fluid will change over time. They want to know if that is true. In some cases, the equations will produce precise values for the state of the fluid at any given moment, only for one of those values to suddenly increase in value, and cause the fluid to go to zero. The equations are said to give rise to a "singularity" at that time.

The equations won't be able to calculate the fluid's flow once they hit that point. Charlie Fefferman said that what people were able to do fell far short ofproving blowup.

It gets more difficult if you want to model a fluid that has a certain amount of elasticity. The Clay Mathematics Institute is offering a million-dollar prize if anyone can prove that the same failures happen in the same equations.