陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「A quantitative formulation of the global regularity problem for the periodic Navier-Stokes」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

It is easy to see that Conjecture 3 implies Conjecture 2, which implies Conjecture 1. By using the compactness of the local periodic Navier-Stokes flow in , one can show that Conjecture 1 implies Conjecture 2; and by using the energy identity (and in particular the fact that the energy dissipation is bounded) one can deduce Conjecture 3 from Conjecture 2. The argument uses only standard tools and is likely to generalise in a number of ways, which I discuss in the paper. (In p

When I previously discussed the Navier-Stokes equations , I suggested that perhaps the best hope to attack this equation was by what I called “Strategy 1”: by obtaining a new a priori bound on solutions to this equation. What this result indicates (in the periodic case) is that this strategy is in fact essentially the only strategy for solving this equation, since the regularity problem is in fact equivalent to that of obtaining an a priori bound.

已知结果和反例

As the qualitative result is now logically equivalent to a quantitative one, it seems to me that purely “soft” approaches to the problem are now extremely unlikely to work, and that a substantial amount of “hard analysis” would have to go into any putative proof of this problem. In particular, it is clear that if one attempts to construct solutions by expressing them as the limit of some sort of regularised (or discretised) solutions, this can only work if one can obtain a pr

证明或构造的主线

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

阅读时建议盯住的点

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

值得单独记下的条目

• (Qualitative regularity conjecture) Given any smooth divergence-free data , there exists a global smooth solution to the Navier-Stokes equations.
• (Local-in-time quantitative regularity conjecture) Given any smooth solution to the Navier-Stokes equations with , one has the a priori bound for some non-decreasing function .
• (Global-in-time quantitative regularity conjecture) This is the same conjecture as 2, but with the condition replaced by .

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源