Navier–Stokes Equations
A system of fundamental partial differential equations in fluid mechanics describing the motion of liquids and gases. The question of the global existence and smoothness of three-dimensional solutions is one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute and remains unresolved.
Core Facts
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?
The 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.
The Navier-Stokes equations are the fundamental partial differential equations describing fluid motion.
Timeline (last 90 days)
A tweet from the social platform X states that last month a large AI agent team consumed more than 100 billion tokens in an attempt to solve the existence and smoothness problem of the Navier-Stokes equations.
The core of the existence and smoothness problem of the Navier-Stokes equations is: under arbitrary initial conditions, do the three-dimensional Navier-Stokes equations always produce smooth, physically reasonable solutions, or can singularities (velocity fields tending to infinity) develop in finite time.
The tweet did not disclose the specific technical architecture or verification results of the AI agent cluster's solving of the Navier-Stokes problem, so it cannot be confirmed whether it truly "solved" the problem.
Alex revealed that within one month he was stunned by AI capabilities for the third time; the first two were the formalization of Fermat's Last Theorem and progress on the Navier-Stokes equations.
The general form of the Navier-Stokes equations is ρ(∂_t u + u·∇u) = -∇p + μ∇²u + f, together with the incompressibility condition ∇·u=0.
The paper points out that in OpenAI's Navier-Stokes proof, equation 8.19 claims that a certain quantity can be bounded by the norm of the (m+4)-th order derivative, but the Lean code verifies that it is bounded by the norm of the (m+5)-th order derivative, which is a weaker conclusion.
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.
GPU fluid simulation typically requires solving the Navier-Stokes equations.
Terence Tao suggests that if a similar self-replicating fluid machine could be constructed in the true three-dimensional Navier-Stokes equation, the answer to the Millennium Prize Problem would most likely be that blowup exists.
Any proof method that relies solely on abstract properties such as energy conservation and scaling analysis cannot prove the global regularity of the Navier-Stokes equation, because the averaged equation satisfies these properties yet still blows up.
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All Facts (20)
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?
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 Navier-Stokes equations are the fundamental partial differential equations describing fluid motion.
80%VerifiedThe Navier-Stokes equations are one of the seven Millennium Prize Problems established by the Clay Mathematics Institute, each carrying a one-million-dollar prize.
75%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 were first proposed by French engineer Claude-Louis Navier in 1822 and later refined by British mathematician George Stokes in 1845.
75%VerifiedThe Navier-Stokes Equations describe the fundamental laws governing the motion of viscous fluids.
75%VerifiedOpenAI announced that approximately 10,000 AI agents spent roughly 88 hours finding a proof related to the Navier-Stokes equations problem.
80%UnverifiedIn mathematics, blowup refers to the L∞ norm of the velocity field tending to infinity at some finite time T*
70%UnverifiedThe tweet did not disclose the specific technical architecture or verification results of the AI agent cluster's solving of the Navier-Stokes problem, so it cannot be confirmed whether it truly "solved" the problem.
50%UnverifiedA tweet from the social platform X states that last month a large AI agent team consumed more than 100 billion tokens in an attempt to solve the existence and smoothness problem of the Navier-Stokes equations.
50%UnverifiedThe core of the existence and smoothness problem of the Navier-Stokes equations is: under arbitrary initial conditions, do the three-dimensional Navier-Stokes equations always produce smooth, physically reasonable solutions, or can singularities (velocity fields tending to infinity) develop in finite time.
50%UnverifiedThe general form of the Navier-Stokes equations is ρ(∂_t u + u·∇u) = -∇p + μ∇²u + f, together with the incompressibility condition ∇·u=0.
50%UnverifiedThe paper points out that in OpenAI's Navier-Stokes proof, equation 8.19 claims that a certain quantity can be bounded by the norm of the (m+4)-th order derivative, but the Lean code verifies that it is bounded by the norm of the (m+5)-th order derivative, which is a weaker conclusion.
50%UnverifiedOpenAI'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.
50%UnverifiedGPU fluid simulation typically requires solving the Navier-Stokes equations.
50%UnverifiedTerence Tao constructed an averaged version of the three-dimensional Navier-Stokes equation in the paper "Finite time blowup for an averaged three-dimensional Navier-Stokes equation".
50%UnverifiedTerence Tao rigorously proved that solutions to the averaged version of the three-dimensional Navier-Stokes equation blow up in finite time.
50%UnverifiedThe core of the Navier-Stokes problem is that the nonlinear convection term concentrates energy into smaller scales and may cause blowup, while the viscous dissipation term tends to dissipate energy and maintain smoothness.
50%UnverifiedThe existing a priori estimates for the three-dimensional Navier-Stokes problem are in a supercritical regime, and the known conserved quantities are insufficient to control the behavior of solutions at small scales.
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