## One regularity of NS 3-D

**Thm:** If solves NS 3-D and , then is regular.

*Proof:* Denote

Note that for all .

Fix , which will be determined later, letting , by Chebyshev’s inequality

Since , then is compact in , in particular, is uniformly integrable, namely

Multiplying the NS equation by , we get

Choose such that small enough and use the Sobolev inequality ,

By Young’s inequality,

So . From the regularity theory, must be a regular solution.

**Remark:** Follows from my note of lectures given by peter constantin.

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