Navier–Stokes Equations

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 describe the fundamental laws governing the motion of viscous fluids.

OpenAI announced that approximately 10,000 AI agents spent roughly 88 hours finding a proof related to the Navier-Stokes equations problem.

In mathematics, blowup refers to the L∞ norm of the velocity field tending to infinity at some finite time T*

The Navier-Stokes equations are one of the seven Millennium Prize Problems established by the Clay Mathematics Institute, each carrying a one-million-dollar prize.

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.

The 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.

The 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.

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.

Terence 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".

Terence Tao rigorously proved that solutions to the averaged version of the three-dimensional Navier-Stokes equation blow up in finite time.

AI has not solved the mathematical challenge of Navier-Stokes, but has provided a new computational paradigm for specific engineering scenarios

Throughout history, claims that the Navier-Stokes problem had been solved have appeared multiple times, but the vast majority were ultimately found to contain critical errors

In the two-dimensional case, mathematicians proved in the 1960s that solutions to the Navier-Stokes equations remain smooth forever

The inclusion of a Lean 4 proof does not mean that OpenAI has solved a Millennium Prize Problem; it more likely targets a specific lemma or special case.

Weather and climate simulation is based on the numerical solving of partial differential equation systems (such as the Navier-Stokes equations), using discretization methods like finite difference and finite element methods, and is extremely sensitive to numerical precision.

OpenAI released a research result related to the Navier-Stokes equations, which includes a Lean 4 formal proof.