We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.
The Millennium Prize Problems (opens in a new window) represent some of the deepest questions at the frontier of mathematics. The question of whether smooth three-dimensional fluid motion can break down has remained unresolved for roughly 90 years.
Really interesting breakdown. The pace of progress in this space is hard to keep up with, and pieces like this help a lot. Thanks for putting it together — bookmarking for reference.
While still doubted regarding agents’ intercommunications, we need to see Navier-Stokes formulation for incompressibles solve several industrial benchmarked problems common in fluid dynamics but intractable in 3D unless those problems are reduced and simplified to 2D or even 1D formulations, with fewer equations and variables. Under AI, i ran several plasma atom vortex formulation using python coded optimized simulation and little-home customized CFDs showed several helically rotational vortices spiraling down to zero irrotational at end-solution.
By the way, proving existence and smoothness of the Navier-Stokes solution while difficult, is less important than attempting to solve real physical 2D and 3D problems without changing
By the way, proving existence and smoothness of the Navier-Stokes solution while difficult, is less important than attempting to solve real physical 2D and 3D problems without simplifying the mathematical modeling of the real physical model.