Long-Sought Proof Tames Some of Math’s Unruliest Equations

To study the flow of air around an airplane’s wing, the distribution of stress on a bridge, or various other situations, researchers use elliptic partial differential equations.

为了研究飞机机翼周围的空气流动、桥梁上的应力分布或各种其他情况，研究人员使用椭圆偏微分方程。

These equations are notoriously difficult to understand.

这些方程非常难以理解。

Kristina Armitage; Michael Kanyongolo/Quanta Magazine

克里斯蒂娜·阿米蒂奇；迈克尔·卡尼永戈洛/Quanta 杂志

The trajectory of a storm, the evolution of stock prices, the spread of disease — mathematicians can describe any phenomenon that changes in time or space using what are known as partial differential equations.

风暴的轨迹、股票价格的演变、疾病的传播——数学家可以使用所谓的偏微分方程来描述任何随时间或空间变化的现象。

But there’s a problem: These “PDEs” are often so complicated that it’s impossible to solve them directly.

但有一个问题：这些“偏微分方程”通常非常复杂，无法直接求解。

Mathematicians instead rely on a clever workaround.

相反，数学家依靠聪明的解决方法。

They might not know how to compute the exact solution to a given equation, but they can try to show that this solution must be “regular,” or well-behaved in a certain sense — that its values won’t suddenly jump in a physically impossible way, for instance.

他们可能不知道如何计算给定方程的精确解，但他们可以尝试证明这个解必须是“规则的”，或者在某种意义上表现良好——例如，它的值不会以物理上不可能的方式突然跳跃。

If a solution is regular, mathematicians can use a variety of tools to approximate it, gaining a better understanding of the phenomenon they want to study.

如果解是正则解，数学家可以使用各种工具来近似它，从而更好地理解他们想要研究的现象。

But many of the PDEs that describe realistic situations have remained out of reach.

但许多描述现实情况的偏微分方程仍然遥不可及。

Mathematicians haven’t been able to show that their solutions are regular.

数学家还无法证明他们的解是正则的。

In particular, some of these out-of-reach equations belong to a special class of PDEs that researchers spent a century developing a theory of — a theory that no one could get to work for this one subclass.

特别是，其中一些遥不可及的方程属于一类特殊的偏微分方程，研究人员花了一个世纪的时间来发展一种理论——一种没有人能够为这个子类工作的理论。

They’d hit a wall.

他们碰壁了。

Now, two Italian mathematicians have finally broken through, extending the theory to cover those messier PDEs.

现在，两位意大利数学家终于取得了突破，将这一理论扩展到了那些更混乱的偏微分方程。

Their paper, published last summer, marks the culmination of an ambitious project that, for the first time, will allow scientists to describe real-life phenomena that have long defied mathematical analysis.

他们去年夏天发表的论文标志着一个雄心勃勃的项目的高潮，该项目将首次让科学家能够描述长期以来无法进行数学分析的现实生活现象。

During a volcanic eruption, a scorching, chaotic river of lava flows over the ground.

火山喷发期间，灼热、混乱的熔岩河从地面上流过。

But after hours or days (or perhaps even longer), it cools enough to enter a state of equilibrium.

但数小时或数天（甚至可能更长时间）后，它冷却到足以进入平衡状态。

Its temperature is no longer changing from moment to moment, although it still varies from place to place across the vast expanse of space the lava covers.

它的温度不再时刻变化，尽管在熔岩覆盖的广阔空间中，各地的温度仍然存在差异。

Mathematicians model systems that change in space but not in time — the temperature of a lava flow at equilibrium, the distribution of nutrients in tissues, the shape of a soap film — using elliptic partial differential equations.

数学家使用椭圆偏微分方程对随空间变化但不随时间变化的系统进行建模——平衡时熔岩流的温度、组织中营养物质的分布、肥皂膜的形状。

From top: Giles Laurent/Creative Commons; Mikael Häggström/Creative Commons; Ted Kinsman/Science Source

从上至下：Giles Laurent/Creative Commons； Mikael Häggström/知识共享；特德·金斯曼/科学来源

Mathematicians describe situations like this using what are called elliptic PDEs.

数学家使用所谓的椭圆偏微分方程来描述这种情况。

These equations represent phenomena that vary across space but not time, such as the pressure of water flowing through rock, the distribution of stress on a bridge, or the diffusion of nutrients in a tumor.

这些方程代表随空间变化但不随时间变化的现象，例如流过岩石的水的压力、桥梁上的应力分布或肿瘤中营养物质的扩散。

But solutions to elliptic PDEs are complicated.

但椭圆偏微分方程的解很复杂。

The solution to the lava PDE, for instance, describes its temperature at every point, given some initial conditions.

例如，熔岩偏微分方程的解描述了在给定一些初始条件的情况下每个点的温度。

It depends on a lot of interacting variables.

它取决于许多相互作用的变量。

Researchers want to approximate such a solution even when it’s impossible to write it down.

研究人员想要近似这样的解决方案，即使不可能写下来。

But the methods they use only work well if the solution is regular — meaning that it doesn’t have any sudden jumps or kinks.

但他们使用的方法只有在解决方案是规则的情况下才有效——这意味着它没有任何突然的跳跃或扭结。

(There won’t be sharp spikes in the lava’s temperature from place to place.) “If something goes wrong, it’s probably because of the [lack of] regularity,” said Makson Santos of the University of Lisbon.

（不同地点的熔岩温度不会出现急剧上升。）“如果出现问题，可能是因为[缺乏]规律，”里斯本大学的马克森桑托斯说。

In the 1930s, the Polish mathematician Juliusz Schauder sought to establish the minimal conditions an elliptic PDE must satisfy to guarantee that its solutions will be regular.

20 世纪 30 年代，波兰数学家 Juliusz Schauder 试图建立椭圆偏微分方程必须满足的最低条件，以保证其解是正则的。

He showed that in many cases, all you have to prove is that the rules baked into the equation — such as the rule for how quickly heat will spread in lava — do not change too abruptly from point to point.

他表明，在许多情况下，你所需要证明的是方程式中的规则——例如热量在熔岩中传播的速度的规则——不会从一个点到另一个点突然改变。

In the decades since Schauder’s proof, mathematicians have shown that this condition is enough to ensure that any PDE that describes a nice, “uniform” material has regular solutions.

自 Schauder 证明以来的几十年里，数学家们已经证明，这个条件足以确保任何描述良好、“均匀”材料的偏微分方程都有正则解。

In such a material, there’s a limit on how extreme the underlying rules can get.

在这种材料中，基本规则的极端程度是有限度的。

For example, if you assume your lava is uniform, heat will always flow within certain speed limits, never too quickly or slowly.

例如，如果您假设熔岩是均匀的，那么热量将始终在一定的速度限制内流动，永远不会太快或太慢。

But lava is actually a diverse mix of molten rock, dissolved gases, and crystals.

但熔岩实际上是熔岩、溶解气体和晶体的多种混合物。

In such a nonuniform material, you can’t control the extremes, and you might get more drastic differences in how quickly heat can spread, depending on your location: Some regions in the lava might conduct heat extremely well, and others extremely poorly.

在这种不均匀的材料中，您无法控制极端情况，并且根据您所在的位置，热量传播的速度可能会出现更大的差异：熔岩中的某些区域可能导热性极好，而另一些区域则导热性极差。

In this case, you’ll use a “nonuniformly elliptic” PDE to describe the situation.

在这种情况下，您将使用“非均匀椭圆”偏微分方程来描述这种情况。

For decades, no one could prove that Schauder’s theory held for this kind of PDE.

几十年来，没有人能够证明肖德的理论对于这种偏微分方程成立。

Unfortunately, “the real world is nonuniformly elliptic,” said Giuseppe Mingione , a mathematician at the University of Parma in Italy.

不幸的是，“现实世界是不均匀的椭圆形”，意大利帕尔马大学数学家朱塞佩·明吉奥内 (Giuseppe Mingione) 说。

That meant mathematicians were stuck.

这意味着数学家陷入了困境。

Mingione wanted to understand why.

Mingione 想要了解原因。

In August 2000, Mingione — 28 years old and fresh off his Ph.D. — found himself in a dilapidated old resort in Russia, attending a conference on differential equations.

2000 年 8 月，28 岁的 Mingione 刚刚获得博士学位。 ——发现自己在俄罗斯一个破旧的老度假村里，参加一个微分方程会议。

One evening, with nothing better to do, he started reading papers by Vasiliĭ Vasil’evich Zhikov, a mathematician he’d met on the trip, and he realized that nonuniformly elliptic PDEs that seem well behaved can have irregular solutions even when they satisfy the condition Schauder had identified.

一天晚上，无事可做，他开始阅读旅途中遇到的数学家瓦西里·瓦西里耶维奇·日科夫 (Vasiliĭ Vasil’evichzhikov) 的论文，他意识到，看似表现良好的非均匀椭圆偏微分方程即使满足肖德确定的条件，也可能有不规则的解。

Schauder’s theory wasn’t simply harder to prove in the nonuniform case.

肖德的理论不仅仅是在非均匀情况下更难证明。

It needed an update.

它需要更新。

Giuseppe Mingione helped prove a conjecture he made 20 years ago.

Giuseppe Mingione 帮助证明了他 20 年前提出的猜想。

The final proof, he said, was “a miracle by desperation.”

他说，最终的证据是“绝望中的奇迹”。

Returning to Italy, he teamed up with two colleagues and proposed that nonuniformly elliptic PDEs needed to satisfy an additional condition to guarantee that their solutions would be regular.

回到意大利后，他与两位同事合作，提出非均匀椭圆偏微分方程需要满足一个附加条件，以保证其解是正则的。

Not only did the rules governing heat flow have to change gradually from point to point, but these changes had to be tightly controlled to account for the lava’s nonuniformity.

不仅控制热流的规则必须逐点逐渐改变，而且必须严格控制这些变化，以解释熔岩的不均匀性。

In particular, the mathematicians posited, the more uneven the material, the tighter this control would have to be.

特别是，数学家们认为，材料越不均匀，这种控制就越严格。

They represented this condition as an inequality, giving a precise threshold for how much nonuniformity a system could tolerate.

他们将这种情况表示为不等式，给出了系统可以容忍多少不均匀性的精确阈值。

They showed that for PDEs where the inequality did not hold, they could no longer guarantee that the solutions would be regular.

他们表明，对于不等式不成立的偏微分方程，他们无法再保证解是正则的。

But they couldn’t prove that the inequality precisely marked the point where solutions would go from being regular to potentially irregular.

但他们无法证明不平等精确地标志着解决方案从常规到潜在不规则的转折点。

Mingione spent years on the problem, to no avail.

Mingione 花了很多年的时间来解决这个问题，但毫无结果。

He eventually abandoned the effort.

他最终放弃了这一努力。

Almost 20 years passed.

快20年过去了。

Then in 2017, a first-year graduate student named Cristiana De Filippis heard about the quest to extend Schauder’s theory to nonuniformly elliptic equations.

然后在 2017 年，一位名叫 Cristiana De Filippis 的一年级研究生听说了将 Schauder 的理论扩展到非均匀椭圆方程的探索。

More experienced mathematicians warned her against pursuing the problem, but she ignored their advice and reached out to Mingione.

更有经验的数学家警告她不要继续研究这个问题，但她忽略了他们的建议并联系了 Mingione。

Over a late-night Skype call, she told him that she had some ideas for how to prove his conjecture and was determined to pick up where he had left off.

在深夜的 Skype 通话中，她告诉他，她对如何证明他的猜想有一些想法，并决心从他停下来的地方继续。

Cristiana De Filippis has been developing a broad theory to better understand the solutions to partial differential equations, setting her sights on more and more complicated cases.

Cristiana De Filippis 一直在发展一种广泛的理论，以更好地理解偏微分方程的解，并将目光投向越来越复杂的情况。

“It was like a time machine,” Mingione said.

“这就像一台时间机器，”明吉奥内说。

“It was like meeting myself of 20 years before and knocking at the door of my own mind.”

“这就像遇见了20年前的自己，敲开了自己的心灵之门。”

According to him, it was De Filippis’ “new energy and enthusiasm and faith that this could be done” that persuaded him to revive his long-dormant attempt to prove his conjecture.

据他说，正是德菲利皮斯的“新的活力、热情和对这一切可以做到的信念”说服了他重新启动长期休眠的尝试来证明他的猜想。

The key to proving that the solution to a PDE is regular is to show that it always changes in a controlled way.

证明偏微分方程的解是正则的关键是证明它总是以受控的方式变化。

Mathematicians do this by looking at a special function that describes how fast the solution changes at each point.

数学家通过查看一个特殊函数来做到这一点，该函数描述了解在每个点变化的速度。

They want to show that this function, which is called the gradient, can’t get too big.

他们想表明这个称为梯度的函数不能变得太大。

But just as it’s usually impossible to directly compute the solution to a PDE, it’s also usually impossible to calculate its gradient.

但正如通常不可能直接计算偏微分方程的解一样，通常也无法计算其梯度。

The Polish mathematician Juliusz Schauder sought to understand when models of physical systems will provide a nice picture of reality, and when they won’t.

波兰数学家朱利叶斯·肖德 (Juliusz Schauder) 试图了解物理系统模型何时能够提供一幅关于现实的美好图景，何时不能。

Instead, De Filippis and Mingione derived what they called a “ghost equation” from the original PDE, a shadow of what they actually needed.

相反，德菲利皮斯和明吉奥内从原始偏微分方程中推导出了他们所谓的“幽灵方程”，这是他们实际需要的影子。

This was where Mingione had gotten stuck decades earlier.

这就是 Mingione 几十年前陷入困境的地方。

But De Filippis had an idea for how to hone the ghost equation so that it could give a crisper view of the PDE.

但德菲利皮斯有一个想法，可以改进幽灵方程，使其能够更清晰地了解偏微分方程。

Using a long, multistep procedure, the pair was able to gain enough information from the ghost equation to recover the gradient.

通过长时间的多步骤程序，两人能够从幽灵方程中获得足够的信息来恢复梯度。

“It’s kind of far-fetched to do it like this,” said Simon Nowak of Bielefeld University in Germany.

德国比勒费尔德大学的西蒙·诺瓦克 (Simon Nowak) 表示：“这样做有点牵强。”

“But it works, and it’s quite beautiful.”

“但它确实有效，而且非常漂亮。”

Now they had to figure out how to show that the gradient they’d recovered couldn’t get too large.

现在他们必须弄清楚如何证明他们恢复的梯度不会变得太大。

They split it into smaller pieces and proved that each piece couldn’t exceed a specific size.

他们将其分成更小的块，并证明每块不能超过特定的尺寸。

This took an enormous amount of effort: Even a tiny measurement error on a single piece would throw off their estimate of the gradient, taking them away from the threshold they were aiming to prove.

这需要付出巨大的努力：即使是单个部件上的微小测量误差也会影响他们对梯度的估计，使他们远离他们想要证明的阈值。

In a 2022 preprint, they were able to tame all these pieces well enough to show that most nonuniformly elliptic PDEs that satisfy Mingione’s inequality have to have regular solutions.

在 2022 年的预印本中，他们能够很好地驾驭所有这些部分，证明满足 Mingione 不等式的大多数非均匀椭圆偏微分方程必须有正则解。

But some PDEs were still missing.

但一些偏微分方程仍然缺失。

To prove the full conjecture, the mathematicians had to get even better bounds on the sizes of the gradient’s pieces.

为了证明完整的猜想，数学家必须对梯度片段的大小有更好的限制。

There was absolutely no wiggle room.

绝对没有回旋的余地。

This required starting over many times — “a never-ending game,” De Filippis said.

这需要多次重新开始——“一场永无止境的游戏，”德菲利皮斯说。

But eventually, they were able to prove that the threshold Mingione had predicted decades earlier was exactly right .

但最终，他们能够证明明吉奥内几十年前预测的阈值是完全正确的。

It was “a miracle by desperation,” he said.

他说，这是“绝望中的奇迹”。

De Filippis and Mingione haven’t just completed a century-long project.

德菲利皮斯和明吉奥内刚刚完成了一个长达一个世纪的项目。

They’ve also made it possible for mathematicians to study complicated real-life processes that until now had to be modeled using unrealistically simplified equations.

它们还使数学家能够研究复杂的现实生活过程，而迄今为止，这些过程必须使用不切实际的简化方程进行建模。

Researchers are also excited to apply their techniques to understand other kinds of partial differential equations, including ones that change in both space and time.

研究人员还很高兴能够应用他们的技术来理解其他类型的偏微分方程，包括随空间和时间变化的偏微分方程。

“The magical part is that they were bringing all this deep theory under one umbrella and then squeezing out the proof,” said Tuomo Kuusi of the University of Helsinki.

赫尔辛基大学的托莫·库西 (Tuomo Kuusi) 表示：“神奇的是，他们将所有这些深刻的理论都集中在一个保护伞下，然后挤出证据。”

PDEs have always been almost prohibitively difficult to analyze mathematically.

偏微分方程的数学分析一直非常困难。

Now they’ve gotten just a little bit easier.

现在他们变得容易了一些。

Behind them, De Filippis said, “there is an enormous reality” waiting to be explained.

德菲利皮斯说，在他们身后，“有一个巨大的现实”等待着解释。

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《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。

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辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。

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