[QuantaMagazine] 수학자들, 유체 방정식을 비물리적 해로 유도

원문: https://www.quantamagazine.org/mathematicians-coax-fluid-equations-into-nonphysical-solutions-20220502/

Mathematicians Coax Fluid Equations Into Nonphysical Solutions | Quanta Magazine

 

Mathematicians Coax Fluid Equations Into Nonphysical Solutions

By Leila Sloman

May 2, 2022

The famed Navier-Stokes equations can lead to cases where more than one result is possible, but only in an extremely narrow set of situations.

A vortex ring was used to prove the new result.

Jennifer Idol/Stocktrek Images/Science Source

For nearly two centuries, all kinds of researchers interested in how fluids flow have turned to the Navier-Stokes equations. But mathematicians still harbor basic questions about them. Foremost among them: How well do the equations adhere to reality?

약 2세기 동안, 유체의 흐름에 관심을 가진 모든 종류의 연구자들이 나비에-스토크스 방정식으로 전환했습니다. 그러나 수학자들은 여전히 그것에 대한 기본적인 질문을 가지고 있습니다. 그 중에서 가장 중요한 것: 이 방정식들이 현실에 얼마나 잘 부합하는가?

 

A new paper set to appear in the Annals of Mathematics has chipped away at that question, proving that a once-promising class of solutions can contain physics-defying contradictions. The advance is another step toward understanding the discrepancy between Navier-Stokes and the physical world —  a mystery that underlies one of math’s most famous open problems.

수학 연감에 발표될 새로운 논문은 그 질문에 대해 일부 해를 제시했으며, 한때 유망하다고 여겨졌던 해들이 물리학을 거스르는 모순을 포함할 수 있음을 증명했습니다. 이 발전은 나비에-스토크스와 실제 세계 사이의 차이를 이해하는 데 있어 또 다른 단계이며, 수학에서 가장 유명한 미해결 문제 중 하나에 깔려 있는 미스터리입니다.

 

“It’s very impressive,” said Isabelle Gallagher, a mathematician at the École Normale Supérieure in Paris and Université Paris Cité. “I mean, it’s the first time you really have [these] solutions which are not 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.

유체는 그 구성 분자들이 일체로 움직이지 않기 때문에 본질적으로 묘사하기 어렵습니다. 이를 고려하여 나비에-스토크스 방정식은 3차원 공간의 각 지점에 대한 속도와 방향을 지정하는 '속도장'을 사용하여 유체를 묘사합니다. 이 방정식은 시작 속도장이 시간에 따라 어떻게 진화하는지를 설명합니다.

 

The big question that mathematicians want to answer: Will the Navier-Stokes equations always work, for any starting velocity field into the arbitrarily distant future? The issue is considered so important that the Clay Mathematics Institute made it the subject of one of their famed Millennium Prize Problems, each of which carries a $1 million bounty.

학자들이 답하고자 하는 큰 질문은 다음과 같습니다: 나비에-스토크스 방정식은 임의로 먼 미래까지 어떠한 시작 속도장에 대해서도 항상 작동할까요? 이 문제는 클레이 수학 연구소가 백만 달러의 상금을 걸고 그들의 유명한 밀레니엄 문제 중 하나로 삼을 만큼 중요하게 여겨집니다.

 

In particular, mathematicians wonder 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 infinite speed might pop up. This outcome, which mathematicians call blow-up, would deviate from the behavior of a real-life fluid. To claim the $1 million prize, a mathematician would have to either prove that blow-up will never happen, or find an example where it does.

특히, 수학자들은 부드럽게 시작하는 해 — 즉, 속도장이 인접한 점 사이에서 갑작스럽게 변하지 않는다는 의미 — 가 항상 부드럽게 유지될지 궁금해합니다. 어느 시점 후에 무한한 속도를 나타내는 날카로운 급증이 나타날 수 있습니다. 이러한 결과를 수학자들은 '폭발'이라고 부르며, 실제 유체의 행동과는 다를 것입니다. 100만 달러의 상금을 주장하기 위해서는, 수학자는 폭발이 결코 일어나지 않을 것임을 증명하거나, 그것이 일어나는 예를 찾아야 합니다.

 

Even if the equations can blow up, perhaps not all is lost. A secondary question is whether a blown-up fluid will always keep flowing in a well-defined, predictable way. More precisely: Is there only a single solution to the Navier-Stokes equations, no matter the initial conditions?

This feature, called uniqueness, is the 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 Technology Lausanne.

방정식이 폭발할 수 있더라도, 모든 것이 상실되는 것은 아닐지도 모릅니다. 두 번째 질문은 폭발한 유체가 항상 잘 정의되고 예측 가능한 방식으로 흐를 것인가입니다. 더 정확하게 말하자면: 초기 조건에 관계없이 나비에-스토크스 방정식에는 오직 하나의 해만이 존재하는가?
이러한 특징을 '유일성'이라고 하며, 이는 댈러스 알브리튼, 엘리아 브루에, 마리아 콜롬보가 소속된 고등연구소 및 스위스 연방공과대학 로잔의 새 논문의 주제입니다.

 

The non-quantum world works in this way. The laws of physics determine how a system evolves from one moment to the next, with no room for guesswork or randomness. If the Navier-Stokes equations can really describe real-life fluids, their solutions should obey the same rules. “If you don’t have uniqueness, then the model is [probably] incomplete,” said Vladimír Šverák, a professor at the University of Minnesota who was Albritton’s doctoral adviser. “It’s simply not possible to describe fluids by the Navier-Stokes equations as people had thought.”

비양자 세계는 이와 같이 작동합니다. 물리 법칙은 시스템이 한 순간에서 다음 순간으로 어떻게 진화하는지를 결정합니다. 추측이나 무작위성은 전혀 없습니다. 만약 나비에-스토크스 방정식이 실제 유체를 진짜로 묘사할 수 있다면, 그 해들은 동일한 법칙들을 따라야 합니다. '유일성이 없다면, 그 모델은 [아마도] 불완전한 것입니다,'라고 알브리튼의 박사 지도교수였던 미네소타 대학의 블라디미르 쉬베락 교수가 말했습니다. '사람들이 생각했던 것처럼 나비에-스토크스 방정식으로 유체를 묘사하는 것은 단순히 불가능합니다.

 

In 1934, the mathematician Jean Leray discovered a novel class of solutions. These solutions could blow up, but just a little bit. (Technically, parts of the velocity field become infinite, but the 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 could help make sense of what happens after blow-up.

1934년에, 수학자 장 르레이는 새로운 종류의 해를 발견했습니다. 이 해들은 폭발할 수 있지만, 조금만. (기술적으로, 속도장의 일부가 무한대가 되지만, 유체의 총 에너지는 유한하게 남아있습니다.) 르레이는 자신의 비부드러운 해가 무한정 지속될 수 있음을 증명할 수 있었습니다. 이 해들이 유일하기도 하다면, 폭발 후에 무슨 일이 일어나는지 이해하는 데 도움이 될 수 있습니다."

 

The new paper, however, has discouraging news. The three authors show that a single Leray starting point can be consistent with two very different outcomes, meaning their tether to reality is weaker than researchers hoped for.

Mathematicians suspected this about Leray solutions, and the last several years 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.

그러나 새로운 논문은 낙담스러운 소식을 전합니다. 세 저자는 단일 르레이 시작점이 매우 다른 두 가지 결과와 일관될 수 있음을 보여주었는데, 이는 연구자들이 기대했던 것보다 현실과의 연결이 약하다는 것을 의미합니다. 수학자들은 르레이 해에 대해 이런 의심을 했으며, 지난 몇 년 동안 증거가 꾸준히 축적되어 왔습니다. 새로운 결과는 '어떤 면에서는 꼭대기에 올려진 체리와 같았다'고 뉴욕 대학교 쿠랑 연구소의 블라드 비콜 교수가 말했습니다.

 

Albritton, Bruè and Colombo entered the picture in the fall of 2020 when they joined a study group at IAS. The purpose of the group was to read twopapers the mathematician Misha Vishik had posted online in 2018. While the most sought-after answers are about the Navier-Stokes equations in three-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.

알브리튼, 브루에, 콜롬보는 2020년 가을에 IAS에서 연구 그룹에 참여하면서 이 분야에 등장했습니다. 그룹의 목적은 수학자 미샤 비시크가 2018년에 온라인에 게시한 두 논문을 읽는 것이었습니다. 가장 많이 찾는 답변들은 3차원 공간의 나비에-스토크스 방정식에 관한 것이지만, 2차원 버전의 방정식들도 존재합니다. 비시크는 이 2D 방정식의 수정된 버전에서 유일성이 없음을 증명했습니다.

 

Yet two years after Vishik posted the papers, the details of his work were still hard to understand. The seven-person study group met regularly for about six months to work through the papers. “With all of us contributing, we were able to see what was going on,” said Albritton.

비시크가 논문을 게시한 지 2년이 지났음에도 불구하고, 그의 연구 내용의 세부사항은 여전히 이해하기 어려웠습니다. 7명으로 구성된 연구 그룹은 약 6개월 동안 정기적으로 만나 논문을 연구했습니다. '우리 모두가 기여함으로써, 우리는 무슨 일이 일어나고 있는지를 볼 수 있었다'고 알브리튼이 말했습니다.

 

Vishik’s 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. But Vishik’s force was a mathematical construct. It wasn’t smooth, and didn’t represent any particular physical process.

비시크의 증명은 외부 힘을 사용했습니다. 실제 세계에서, 힘은 물이 튀는 것, 바람 또는 유체의 궤적을 변경할 수 있는 다른 어떤 것에 의한 것일 수 있습니다. 그러나 비시크의 힘은 수학적 구조였습니다. 그것은 부드럽지 않았고 특정한 물리적 과정을 대표하지 않았습니다.

 

With that force in place, Vishik had been able to find two distinct solutions to the two-dimensional equations. His solutions were based off of a vortex-like flow.

그 힘이 적용되면서, 비시크는 2차원 방정식에 대해 두 가지 구별되는 해를 찾을 수 있었습니다. 그의 해결책은 소용돌이 같은 흐름을 기반으로 했습니다.

 

 

“It’s essentially creating a fluid flow that’s just swirling you around,” said Albritton.

Albritton and Colombo — later joined by Bruè — realized they could use Vishik’s vortex as the foundation for two distinct solutions in three dimensions as well.

“The strategy is actually very innovative,” said Vicol, who advised Albritton during the latter’s postdoctoral fellowship at NYU.

To prove non-uniqueness, the three authors constructed a doughnut-shaped “vortex ring” solution to the three-dimensional equations. At first, their fluid is completely still, but a force propels it into motion. This force, like Vishik’s, is not smooth, ensuring that the vortex ring will not be smooth either. As the fluid gains momentum, it flows along the vortices, circling through the doughnut hole and back up around the outside.

The authors then showed that this vortex ring solution can degenerate into a different 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 in the Navier-Stokes equations as a “perturbation” added to the velocity field. You can play with the size of that perturbation by dropping the stone more or less gently; if you drop it very carefully from a point close to the surface, it might barely affect the lake at all.

But if you drop a stone into the flow that Albritton, Bruè and Colombo created, the perturbation will never disappear. Even if you drop the stone from effectively zero height, that vanishingly tiny disturbance would grow into something much more formidable. That creates a second distinct solution from the same initial conditions.

“You have one solution, and instead of making a finite disturbance, you make an infinitesimally small disturbance,” said Albritton. “And then, instantly the solutions are driven apart.”

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 would prefer to avoid the addition of a force altogether and prove that some set of initial conditions leads to non-uniqueness without any outside influence. That question is now perhaps a stone’s throw closer to being answered.

Editor’s note: Dallas Albritton has received funding from the Simons Foundation, which also funds this editorially independent magazine.