Flash - At Least One Of Euler's Equations Does Fail!!!!!!

 For the past few centuries, mathematicians have been striving to create equations that model the motion of fluids.  Some of these equations such as the one that models the ripples that cross a pond, have been used to predict the weather, help design better airplanes, and help explain how blood circulates through the body.  

Although the equations can appear quite simple, the answers can be a lot harder to explain, in fact, almost ridiculously difficult. One of the oldest of those equations was formulated over 250 years ago by Leonhard Euler. It described the flow of a fluid with no viscosity, no internal friction, and can't be compressed into a smaller volume.  Practically all of the equations dealing with nonlinear fluids have been derived from this formula.

Unfortunately, no one is sure if the equation actually models ideal fluid flow.  Mathematicians have been looking to see if there are any points at which it fails, where the equation breaks down.  Now there are two mathematicians who have shown that one particular set of his equations sometimes does fail.  It does not solve the problems with the more general version but offers hope.

The 177 page proof is the result of 10 years of computer based research. This does make it hard for other mathematicians to check the proof and it has people looking at the question of what makes a proof along with the idea of how viable is it if the only way to prove something is with the use of a computer.  

As far as Euler's equations, if you know the location and velocity of each and every particles in the fluid, it should be able to predict how the fluid changes and evolves over time but mathematicians want to know if this is true for all cases. If at any time the values shoot up to infinity, this singularity then blows up at that point and fails.  Once that singularity is found, the equation no longer can calculate the fluid's flow.  Everything becomes more complicated if you try to model a fluid with viscosity. 

In addition, it is very hard to prove a singularity of this type because most computers are unable to compute infinite values. A computer is able to get close but cannot compute the actual values so it is not an actual proof. Instead, mathematicians have to go back to a previous point that gives them a self-similar solution.  Two mathematicians came up with a possible point but were unable to prove it so they went back, looked at things and developed a hybrid approach.  

They decided they were proving that if you took any set of values close to the approximate solution and put it into the equation, the results wouldn't be that far off.  So they had to define closeness before they could create a complex inequality using terms from rescaled equations and the approximate solution.  They also had to make sure that everything came out balanced to something small. 

They ended up breaking the inequality into two parts. The first part could be solved by hand using techniques from the 18th century but the second part required the assistance of a computer due to the number of calculations and precision needed. Using computers to help prove in this particular field is relatively new and will take a while before people are able to check their work.  Let me know what you think, I'd love to hear.  Have a great day.