for nearly two centuries, all kinds of researchers interested in how fluids flo ‘ve turned to the navier-stokes equations. but mathematicians still harbor basic ?s bout them. primordial among them: how well do the equations adhere to reality?

a new paper set to appear inna *annals of mathematics* has chipped away at that ?, proving dat a' once-promising class of solutions can contain physics-defying contradictions. the advance is another step toward cogging the discrepancy tween navier-stokes na physical realm — a mystery that underlies one of math’s most famous open problems.

“it’s very impressive,” said isabelle gallagher, a mathematician atta École nor♂ supérieure in paris and université paris cité. “i mean, it’s the 1st time you really ‘ve [these] solutions which aint unique.”

fluids are inherently difficult to describe, as their constituent molecules don’t move as one. to account for this, the navier-stokes equations describe a fluid using “velocity fields” that specify a speed and direction for each point in 3d space. the equations describe how a starting velocity field evolves over time.

the big ? that mathematicians wanna answer: will the navier-stokes equations always work, for any starting velocity field inna'da arbitrarily distant future? the issue is pondered so primordial that the clay mathematics institute made it the subject of one o'their famed millennium prize problems, each of which carries a $1 million bounty.

in pticular, mathematicians wanda whether a solution that starts out smooth — meaning its velocity fields don’t change abruptly from one nearby point to another — will always remain smooth. it’s possible that after a while, sharp spikes that represent ∞ speed mite pop up. this outcome, which mathematicians call blo-up, ‘d deviate from the behavior offa real-life fluid. to claim the $1 million prize, a mathematician ‘d ‘ve to either prove that blo-up will never happen, or find an ex where t'does.

even if the equations can blo up, perhaps not all is lost. a 2ndary ? is whether a blon-up fluid will always keep floing in a well-defined, predictable way. + precisely: is there 1-ly a single solution to the navier-stokes equations, no matter the initial conditions?

this feature, called uniqueness, tis subject of the new paper by dallas albritton and elia bruè of the institute for advanced study and maria colombo of the swiss federal institute of tek lausanne.

the non-quantum realm wox'n this way. the laws of physics determine how a system evolves from one moment to the nxt, with no room for guesswork or randomness. if the navier-stokes equations can really describe real-life fluids, their solutions ‘d obey the same rules. “if you don’t ‘ve uniqueness, then the model is [probably] incomplete,” said vladimír Šverák, a professor atta university of minnesota who was albritton’s dral adviser. “it’s simply not possible to describe fluids by the navier-stokes equations as pplz had thought.”

in 1934, the mathematician jean leray discovered a novel class of solutions. these solutions ‘d blo up, but just a lil bit. (teknically, pts of the velocity field become ∞, but'a fluid’s total energy remains finite.) leray was able to prove that his non-smooth solutions can go on indefinitely. if these solutions are also unique, then they ‘d help make sense of wha’ happens after blo-up.

the new paper, however, has discouraging news. the 3 authors show dat a' single leray starting point can be consistent with two very ≠ outcomes, meaning their tether to reality is weaker than researchers hoped for.

mathematicians suspected this bout leray solutions, na last several yrs saw a steady accumulation of evidence. the new result “was somehow the cherry on top,” said vlad vicol, a professor at new york university’s courant institute.

albritton, bruè and colombo entered the picture inna fall of 2020 when they joined a study group at ias. the purpose of the group was to read two papers the mathematician misha vishik had posted online in 2018. while the most sought-after answers are bout the navier-stokes equations in 3-dimensional space, two-dimensional versions of the equations also exist. vishik had proved that non-uniqueness occurs in a modified version of these 2d equations.

yet two yrs after vishik posted the papers, the details of his work were still hard to cogg. the 7-person study group met regularly for bout 6 mnths t'work through the papers. “with all of us contributing, we were able to see wha’ was goin on,” said albritton.

vishik’s proof used an external force. in a real-realm setting, a force mite be due to splashing, wind, or anything else w'da ability to change a fluid’s trajectory. but vishik’s force was a mathematical construct. twasn’t smooth, and didn’t represent any pticular physical process.

with that force in place, vishik had been able to find two distinct solutions to the two-dimensional equations. his solutions were based off offa vortex-like flo.

“it’s primordially creating a fluid flo that’s just swirling you round,” said albritton.

albritton and colombo — l8r joined by bruè — realized they ‘d use vishik’s vortex as the foundation for two distinct solutions in 3 dimensions swell.

“the strategy is actually very innovative,” said vicol, who advised albritton during the latter’s postdral felloship at nyu.

to prove non-uniqueness, the 3 authors constructed a doughnut-shaped “vortex ring” solution to the 3-dimensional equations. at 1st, their fluid is completely still, but a force propels it into motion. this force, like vishik’s, aint smooth, ensuring that the vortex ring will not be smooth either. as the fluid gains momentum, it flos along the vortices, circling through the doughnut hole and back up round the outside.

the authors then showed that this vortex ring solution can degenerate into a ≠ solution.

the effect was something like dropping a stone into a lake. typically, you’ll see a few waves that dissipate after a short time. those waves show up inna navier-stokes equations as a “perturbation” added to the velocity field. you can play w'da size odat perturbation by dropping the stone + or less gently; if you drop it very carefully from a point close to the surface, it mite barely affect the lake at all.

but if you drop a stone inna'da flo that albritton, bruè and colombo created, the perturbation will never disappear. even if you drop the stone from effectively zero h8, that vanishingly tiny disturbance ‘d grow into something much + formidable. that creates a 2nd distinct solution from the same initial conditions.

“you ‘ve one solution, and instead of making a finite disturbance, you make an ∞simally lil disturbance,” said albritton. “and then, instantly the solutions are driven apt.”

the new paper does not definitively settle whether leray solutions are unique. its conclusions rely on an external force crafted specifically to make non-uniqueness occur. mathematicians ‘d prefer to avoid the addition offa force altogether and prove that some set of initial conditions leads to non-uniqueness without any outside influence. that ? is now perhaps a stone’s throw closer to bein’ answered.

*editor’s note: dallas albritton has received funding from the simons foundation, which also funds this* *editorially indie magazine**.*

original content at: www.quantamagazine.org…

authors: leila sloman

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