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「Conserved quantities for the Euler equations」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

The Euler equations for incompressible inviscid fluids may be written as

where is the velocity field, and is the pressure field. To avoid technicalities 下面会 assume that both fields are smooth, and that is bounded. We will take the dimension to be at least two, with the three-dimensional case being of course especially interesting.

已知结果和反例

The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post , one potential route to establishing finite time blowup for the latter equations when is to be able to construct “computers” solving the Euler equations, which generate smaller replicas of themselves in a noise-tolerant manner (as the viscosity term in the Navier-Stokes equation is to be viewed as perturbative noise).

Perhaps the most prominent obstacles to this route are the conservation laws for the Euler equations, which limit the types of final states that a putative computer could reach from a given initial state. Most famously, we have the conservation of energy

证明或构造的主线

(assuming sufficient decay of the velocity field at infinity); thus for instance it would not be possible for a computer to generate a replica of itself which had greater total energy than the initial computer. This by itself is not a fatal obstruction (in this paper of mine , I constructed such a “computer” for an averaged Euler equation that still obeyed energy conservation). However, there are other conservation laws also, for instance in three dimensions one also has cons

and (formally, at least) one has conservation of momentum

阅读时建议盯住的点

(although, as we shall discuss below, due to the slow decay of at infinity, these integrals have to either be interpreted in a principal value sense, or else replaced with their vorticity-based formulations, namely impulse and moment of impulse). Total vorticity

is also conserved, although it turns out in three dimensions that this quantity vanishes when one assumes sufficient decay at infinity. Then there are the pointwise conservation laws: the vorticity and the volume form are both transported by the fluid flow, while the velocity field (when viewed as a covector) is transported up to a gradient; among other things, this gives the transport of vortex lines as well as Kelvin’s circulation theorem , and can also be used to deduce th

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