Researchers have been interested in how fluids flow for nearly two centuries. Basic questions about mathematicians still exist. How well do the equations follow reality?
A new paper set to appear in the Annals of Mathematics proves that a once promising class of solutions can contain physics-defying contradictions. The discrepancy between the physical world and the Navier-Stokes is a mystery that underlies one of math's most famous open problems.
The mathematician at the cole Normale Supérieure in Paris said it was very impressive.
It's difficult to describe fluids as they don't move as one. To account for this, the equations describe a fluid usingvelocity fields that specify a speed and direction for each point. The equations show how the field changes over time.
The big question that mathematicians want to answer is: Will the equations always work for a starting field into a distant future? The Clay Mathematics Institute made the issue the subject of one of their famed Millennium Prize Problems, each of which carries a $1 million bounty.
The mathematicians wonder if a solution that starts out smooth will not change abruptly from one point to another. It is1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556 The blow-up outcome would be different from a real-life fluid. To win the $1 million prize, a mathematician has to prove that blow-up will never happen or find an example where it does.
Maybe not all is lost even if the equations blow up. There is a question of whether a blown-up fluid will always flow in a predictable way. Is there a single solution to the equations no matter what the initial conditions are?
The paper by Dallas Albritton and Elia BruxE8; of the Institute for Advanced Study and Maria Colombo of the Swiss Federal Institute of Technology Lausanne is about this feature.
This is how the non-quantum world works. The laws of physics determine how a system can evolve from one moment to the next. The solutions should obey the same rules if the equations can describe real-life fluids.
Jean Leray discovered a new class of solutions in 1934. These solutions could blow up. The fluid's total energy remains finite, even though parts of the velocity field become infinite. Leray was able to show that his solutions can go on indefinitely. The solutions could help make sense of what happens after a blow-up.
The new paper has bad news. Three authors show that a single Leray starting point can be consistent with two very different outcomes, meaning their tether to reality is not as strong as they should be.
The last several years have seen a steady accumulating of evidence. The new result was the cherry on top, according to a professor at New York University.
The fall of 2020 saw Albritton, BruxE8; and Colombo join a study group. The two papers that the group was to read were posted on the internet. There are two-dimensional versions of the equations that exist. In a modified version of these 2D equations, non-uniqueness occurs.
Two years after the papers were posted, the details of his work were still hard to comprehend. The seven people in the study group were able to see what was happening.
The proof used an external force. In a real-world setting, a force might be due to splashing, wind, or anything else with the ability to change a fluid's trajectory. The force was a mathematical construct. It wasn't smooth and didn't represent a physical process.
With that force in place, Vishik was able to find two different solutions to the two-dimensional equations. His solutions were based on the flow.
Albritton said it was creating a fluid flow that was swirling around.
The foundation for two different solutions in three dimensions was realized by Albritton and Colombo.
The strategy is very innovative, according to Vicol, who advised Albritton during his fellowship at NYU.
The solution to the three-dimensional equations was constructed by the three authors. Their fluid is still at first, but a force propels it into motion. The force is not smooth and the ring will not be smooth either. As the fluid gains strength, it flows through the doughnut hole and back up around the outside.
The authors showed that the solution can be different.
It was like dropping a stone into a lake. You will usually see a few waves that go away after a short time. The waves show up in the equations as a perturbation. If you gently drop the stone from a point close to the lake, it will not affect the lake at all.
If you drop a stone into the flow created by Albritton, Bru, and Colombo, it will never go away. If you dropped the stone from zero height, it would grow into something much more formidable. A second solution is created from the same initial conditions.
The solutions are driven apart when you make an infinitesimally small disturbance.
The paper doesn't say whether Leray solutions are unique. It relies on an external force to make it happen. Mathematicians would prefer to avoid the addition of a force altogether and prove that some initial conditions lead to non-uniqueness without any outside influence. The question is throwing closer to being answered.
Dallas Albritton has received funding from the Simons Foundation, which also funds this editorially independent magazine.
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