Now one group of mathematicians has developed a way of training machines to spot these phantom glitches. In a preprint posted in September, they reexamined simpler fluid equations already known to host a stable singularity. There, they found additional potential blowup scenarios<\/a> \u2014 including unstable ones. It was the first time a possible unstable singularity was uncovered in a fluid of more than one dimension.<\/p>\n The team went on to discover an assortment of unstable singularity candidates in several other fluid equations as well. They have not found any million-dollar singularities. And they still need to rigorously prove that the ones they have found do indeed blow up. But their success in uncovering potential unstable singularities in simple models raises hopes that it will also be possible to find unstable blowups in higher-stakes scenarios.<\/p>\n \u201cThe idea of an unstable singularity no longer prevents the discovery of the singularity,\u201d said Fefferman, who was not involved in the new research.<\/p>\n Singularity Hunting<\/p>\n A solution to the Navier-Stokes equations captures a slice of eternity. Solving the equations for some initial state of the fluid will tell you the fluid\u2019s velocity at each point in space and at every moment in time. In one simple solution, a fluid might start calm and remain calm forever. In a more complicated setup, gentle currents might merge into whirlpools and eddies. The great mystery is whether every solution \u2014 every single possible fluid history that satisfies the Navier-Stokes equations \u2014 makes sense everywhere and always.<\/p>\n But tackling the Navier-Stokes equations for fluids in three dimensions is unspeakably difficult, so mathematicians have started with easier versions of the problem. For instance, the Euler equations assume that fluids flow with no internal friction, or viscosity. Energy doesn\u2019t dissipate in these frictionless fluids, so they should blow up more easily than viscous ones.<\/p>\n But even in this simpler scenario, finding a blowup solution is hard. Fluid equations are generally too complicated to solve directly with pencil and paper. So a common approach is to use a computer to simulate the fluid\u2019s motion and get an approximate sense of the conditions that seem to produce a blowup. If you can narrowly identify the blowup-producing conditions, you might be able to use that knowledge to rigorously prove that a blowup truly exists.<\/p>\n That\u2019s the approach that Thomas Hou<\/a> and Guo Luo<\/a> took in 2013, when they simulated a digital liquid in a can. They set the top half of the liquid spinning in one direction and the bottom half in the other, then evolved this scenario through time using the Euler equations. Eventually, at points where the opposing flows met along the can\u2019s boundary, the vorticity (a measure of how much the liquid spins around a point) got big \u2014 bigger than their computer could handle.<\/p>\n Mark Belan\/Quanta Magazine<\/p>\n This was a hint that a similar set of conditions would lead to a blowup. But it was not a guarantee. \u201cThe graveyards are strewn with alleged singular solutions of 3D Euler,\u201d Fefferman said.<\/p>\n It took Hou and another collaborator, Jiajie Chen<\/a>, nearly a decade to remove the \u201calleged.\u201d In 2022, they used a computer to prove that the singularity candidate implied the existence of a true singularity<\/a>. It was a landmark proof, and it got mathematicians hungry to push the frontier even further.<\/p>\n The research depended on computer simulations, which meant that tiny adjustments to the initial state of the digital fluid (or any digital rounding errors) wouldn\u2019t affect the fluid\u2019s fate. A singularity would still occur at the can\u2019s boundary even if things played out a little bit differently.<\/p>\n Because of this, the singularity was stable. But a singularity need not be stable. A blowup might occur only when the fluid is set up in the most delicate of ways. In such a case, any adjustment to that initial arrangement, no matter how small, would prevent the fluid from blowing up.<\/p>\n Many mathematicians conjecture that if singularities do lurk in more realistic fluid equations, they\u2019ll be unstable like this, springing up without warning.<\/p>\n They\u2019ll also be far harder to find.<\/p>\n Going Finite<\/p>\n It\u2019s essentially impossible to track down an unstable singularity candidate with a computer simulation. First you\u2019d need a cosmic stroke of luck to land on exactly the right initial configuration for your fluid \u2014 akin to trying to balance a pen perfectly on its tip, said Tristan Buckmaster<\/a>, a mathematician at New York University. Then, to keep it balanced, you\u2019d also have to evolve the fluid flawlessly from one moment to the next, since even the smallest deviation will tip it onto a path that doesn\u2019t blow up.<\/p>\n Computers aren\u2019t capable of infinite precision. They\u2019ll inevitably introduce numerical errors that, though tiny, will stop the unstable singularity from forming. \u201cIt\u2019s like the wind blowing on your pen,\u201d Buckmaster said.<\/p>\n","protected":false},"excerpt":{"rendered":"Nearly 200 years ago, the physicists Claude-Louis Navier and George Gabriel Stokes put the finishing touches on a…\n","protected":false},"author":2,"featured_media":229592,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20],"tags":[345,343,344,85,46,125],"class_list":{"0":"post-229591","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-artificial-intelligence","8":"tag-ai","9":"tag-artificial-intelligence","10":"tag-artificialintelligence","11":"tag-il","12":"tag-israel","13":"tag-technology"},"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/posts\/229591","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/comments?post=229591"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/posts\/229591\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/media\/229592"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/media?parent=229591"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/categories?post=229591"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/tags?post=229591"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/www.quantamagazine.org\/wp-content\/uploads\/2026\/01\/Unstable_Blowup-Fig1-crMarkBelan-Desktopv1-1.svg\")
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