Navier–Stokes Equations
A system of partial differential equations describing the motion of viscous fluids and constituting the core equations of fluid dynamics. The existence and smoothness of their solutions is one of the seven Millennium Prize Problems established by the Clay Mathematics Institute.
Core Facts
The Millennium Prize Problems were established by the Clay Mathematics Institute in 2000, comprising seven mathematical problems, each carrying a prize of one million US dollars upon solution.
The Navier-Stokes equations were independently derived in the 19th century by French engineer Claude-Louis Navier and British mathematician George Stokes.
The core question of the Navier-Stokes Millennium Prize Problem is: given smooth initial conditions in three-dimensional space, does the equation always have a global smooth solution?
Timeline (last 90 days)
Solving the Navier-Stokes equations one month ago involved approximately 10,000 agents and took 88 hours.
Generative models do not guarantee energy conservation, do not guarantee that fluids satisfy the Navier-Stokes equations, and only guarantee that the output is statistically close to the training data in visual terms.
OpenAI used thousands of agents to produce a proof attempt for the existence and smoothness problem of the Navier-Stokes equations within 88 hours.
OpenAI's previous work on the Navier–Stokes equations was a formal verification of an existing mathematical proof of that equation, not the discovery of a new proof from scratch.
A discussion thread titled "Did OpenAI solve the wrong Navier-Stokes problem?" appeared on Hacker News, questioning whether OpenAI solved the wrong Navier-Stokes problem.
The true Millennium Prize Problem refers to a universal proof of whether smooth solutions always exist for three-dimensional incompressible fluids under general conditions — a question that has neither been proven nor disproven to this day.
When faced with headlines claiming AI has solved a world-class problem, asking which specific version of the problem was addressed is an effective method for evaluating AI research results.
In the Navier-Stokes problem, the three-dimensional problem can be partially solved after adding additional geometric assumptions such as axisymmetric irrotational flow.
The core question in the discussion is not whether the answer given by the AI is correct, but rather which version of the problem the AI was actually solving.
A YouTube video claims that OpenAI's AI system has solved the Navier-Stokes equations problem.
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All Facts (20)
The Millennium Prize Problems were established by the Clay Mathematics Institute in 2000, comprising seven mathematical problems, each carrying a prize of one million US dollars upon solution.
90%VerifiedThe Navier-Stokes equations were independently derived in the 19th century by French engineer Claude-Louis Navier and British mathematician George Stokes.
90%VerifiedThe core question of the Navier-Stokes Millennium Prize Problem is: given smooth initial conditions in three-dimensional space, does the equation always have a global smooth solution?
90%VerifiedThe existence and smoothness problem of the Navier-Stokes equations is one of the seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000, each carrying a $1 million prize.
90%VerifiedThe Millennium Prize Problems were established by the Clay Mathematics Institute in 2000, comprising seven problems each carrying a prize of $1 million.
80%VerifiedThe two-dimensional version of the Navier-Stokes equations has been proven to have global smooth solutions, but the three-dimensional case remains unsolved due to the vorticity stretching mechanism.
75%VerifiedThe Navier-Stokes Equations describe the fundamental laws governing the motion of viscous fluids.
75%VerifiedThe Navier-Stokes equations are a system of partial differential equations describing fluid motion and are a core tool in fluid mechanics.
65%VerifiedThe existence of smooth solutions to the Navier–Stokes equations in three-dimensional space has not yet been mathematically proven, and is one of the Millennium Prize Problems offered by the Clay Mathematics Institute.
65%VerifiedOpenAI announced that approximately 10,000 AI agents spent roughly 88 hours finding a proof related to the Navier-Stokes equations problem.
80%Unverified克雷数学研究所设立的七个千禧年大奖难题每题奖金百万美元，目前仅黎曼猜想等六题仍未解决
85%Partially VerifiedThe prize money for the Navier-Stokes Millennium Prize Problem is $1,000,000
70%UnverifiedThe three-dimensional vorticity equation contains a vortex stretching term through which energy may rapidly concentrate toward small scales, while two-dimensional fluids lack this mechanism—this is why the three-dimensional case is fundamentally more difficult.
70%UnverifiedThe Navier-Stokes equations were first proposed by French engineer Claude-Louis Navier in 1822 and later refined by British mathematician George Stokes in 1845.
70%Partially VerifiedThe Navier-Stokes equations are the fundamental partial differential equations describing fluid motion.
65%UnverifiedThe Navier-Stokes equations are one of the seven Millennium Prize Problems established by the Clay Mathematics Institute, each carrying a one-million-dollar prize.
60%UnverifiedBuckmaster and Vlad Vicol proved the non-uniqueness of weak solutions to the Navier-Stokes equations based on the Convex Integration technique.
60%UnverifiedThe Navier-Stokes equations describe the fundamental laws governing viscous fluid motion, and the existence and smoothness of their mathematical solutions is one of the seven Millennium Prize Problems of the Clay Mathematics Institute
60%UnverifiedSolving the Navier-Stokes equations one month ago involved approximately 10,000 agents and took 88 hours.
50%UnverifiedGenerative models do not guarantee energy conservation, do not guarantee that fluids satisfy the Navier-Stokes equations, and only guarantee that the output is statistically close to the training data in visual terms.
50%