[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"project-96093":3},{"id":4,"name":5,"fullName":6,"owner":7,"repo":5,"description":8,"homepage":9,"htmlUrl":10,"language":11,"languages":10,"totalLinesOfCode":10,"stars":12,"forks":13,"watchers":14,"openIssues":15,"contributorsCount":15,"subscribersCount":15,"size":15,"stars1d":15,"stars7d":15,"stars30d":16,"stars90d":15,"forks30d":15,"starsTrendScore":15,"compositeScore":17,"rankGlobal":10,"rankLanguage":10,"license":18,"archived":19,"fork":19,"defaultBranch":20,"hasWiki":19,"hasPages":19,"topics":21,"createdAt":10,"pushedAt":10,"updatedAt":22,"readmeContent":23,"aiSummary":24,"trendingCount":15,"starSnapshotCount":15,"syncStatus":25,"lastSyncTime":26,"discoverSource":27},96093,"NavierStokesAndEuler","openai\u002FNavierStokesAndEuler","openai","Lean certificates accompanying Navier-Stokes and Euler results","",null,"Lean",1777,177,36,0,315,64.75,"Apache License 2.0",false,"main",[],"2026-09-20 04:01:32","# Finite time blowup for Navier–Stokes and Euler equations\n\nThis repository contains Lean 4 formalizations of the results presented in\n“[Finite time blowup for Navier–Stokes](https:\u002F\u002Fcdn.openai.com\u002Fpdf\u002F32d9f210-8b73-45e0-91bc-82a30aef8a9a\u002Fnavier-stokes.pdf)” and\n“[Finite time blowup for the Euler equation](https:\u002F\u002Fcdn.openai.com\u002Fpdf\u002F315b36cd-ec98-4023-8342-93345194ece1\u002Feuler.pdf)” by OpenAI.\n\n- [Read the blog post](https:\u002F\u002Fopenai.com\u002Findex\u002Fnavier-stokes-solution\u002F)\n- [Read the Navier-Stokes paper](https:\u002F\u002Fcdn.openai.com\u002Fpdf\u002F32d9f210-8b73-45e0-91bc-82a30aef8a9a\u002Fnavier-stokes.pdf)\n- [Read the Euler paper](https:\u002F\u002Fcdn.openai.com\u002Fpdf\u002F315b36cd-ec98-4023-8342-93345194ece1\u002Feuler.pdf)\n\n## Navier Stokes\n\nFor every positive viscosity, we prove two results:\n\n- **Whole space $\\mathbb{R}^3$:** There exist smooth initial data and forcing for\n  which no global smooth solution with uniformly bounded kinetic energy exists.\n- **Periodic torus $\\mathbb{R}^3\u002F\\mathbb{Z}^3$:** There exist smooth periodic\n  initial data and forcing for which no global smooth solution exists.\n\nThese are alternatives [**(C)**](https:\u002F\u002Fwww.claymath.org\u002Fwp-content\u002Fuploads\u002F2022\u002F06\u002Fnavierstokes.pdf#page=2) “Breakdown of Navier–Stokes solutions on ℝ³”\nand [**(D)**](https:\u002F\u002Fwww.claymath.org\u002Fwp-content\u002Fuploads\u002F2022\u002F06\u002Fnavierstokes.pdf#page=2) “Breakdown of Navier–Stokes Solutions on ℝ³\u002Fℤ³”\nin the Clay Mathematics Institute’s [official problem description](https:\u002F\u002Fwww.claymath.org\u002Fwp-content\u002Fuploads\u002F2022\u002F06\u002Fnavierstokes.pdf)\nof the [Navier–Stokes existence and smoothness](https:\u002F\u002Fwww.claymath.org\u002Fmillennium\u002Fnavier-stokes-equation\u002F)\n[Millennium Prize Problem](https:\u002F\u002Fwww.claymath.org\u002Fmillennium-problems\u002F).\n\n## Euler\n\nWe construct smooth, compactly supported, divergence-free initial velocity on\n$\\mathbb{R}^3$ whose solution to the unforced incompressible Euler equations\ndevelops a singularity in finite time. The velocity’s $C^1$ norm becomes unbounded\nnear that time, and the time integral of the vorticity’s $L^\\infty$ norm diverges.\n\n## Building the formalizations\n\nThe project uses Lean 4.34.0-rc2, Mathlib, and Lake. With\n[elan](https:\u002F\u002Fgithub.com\u002Fleanprover\u002Felan) installed, fetch the mathlib cache and build the formalizations with:\n\n```sh\nlake exe cache get\nlake build\n```\n\n## Independent proof checking\n\nFor instructions on checking the formalizations with Comparator, see the\n[ComparatorChallenges README](ComparatorChallenges\u002FREADME.md).\n","该项目是OpenAI发布的Navier-Stokes方程与Euler方程有限时间奇点（blowup）结果的Lean 4形式化验证库。核心功能是使用Lean定理证明器对两篇关键论文中关于不可压流体方程解破裂的严格数学证明进行机器可验证的形式化编码，覆盖全空间与环面两类Navier-Stokes设定，以及三维Euler方程的奇点构造。技术特点包括基于Mathlib的高阶数学库支持、端到端可独立验证的证明结构，以及与Comparator工具链兼容的验证流程。适用于数学基础、偏微分方程理论验证、形式化方法在分析学中的应用等研究场景。",2,"2026-09-10 02:30:03","CREATED_QUERY"]