We prove that for a given smooth initial value, if a finite element solution of the three-dimensional Navier–Stokes equations is bounded in a certain norm with a relatively small mesh size, then the solution of the Navier–Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical solution.

中文翻译：

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网格尺寸小的有界数值解意味着存在Navier-Stokes方程的光滑解

我们证明，对于给定的平滑初始值，如果三维Navier–Stokes方程的有限元解以某个相对较小的网格尺寸约束在某个范数内，则带有给定初始值的Navier–Stokes方程的解值必须是光滑且唯一的，并且可以通过数值解成功近似。