For more than 250 years, mathematicians have wondered if the Euler equations might sometimes fail to describe a fluid’s flow. Now there’s a breakthrough.
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Perhaps the oldest and most prominent of these equations, formulated by Leonhard Euler more than 250 years ago, describe the flow of an ideal, incompressible fluid: a fluid with no viscosity, or internal friction, and that cannot be forced into a smaller volume. “Almost all nonlinear fluid equations are kind of derived from the Euler equations,” saidYet much remains unknown about the Euler equations—including whether they’re always an accurate model of ideal fluid flow.
Mathematicians have long suspected that there exist initial conditions that cause the equations to break down. But they haven’t been able to prove it.posted online in October, a pair of mathematicians has shown that a particular version of the Euler equations does indeed sometimes fail. The proof marks a major breakthrough—and while it doesn’t completely solve the problem for the more general version of the equations, it offers hope that such a solution is finally within reach.
, a mathematician at the University of Maryland who was not involved in the work. “There are no results of its kind in the literature.”The 177-page proof—the result of a decade-long research program—makes significant use of computers. This arguably makes it difficult for other mathematicians to verify it.
Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow. But “as of a few years ago, what people were able to do fell very, very far short of [proving blowup],” saidIt gets even more complicated if you’re trying to model a fluid that has viscosity .
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