关于3D不可压Navier-Stokes方程H¹正则性的注记

A Note on the H¹ Regularity of 3D Incompressible Navier-Stokes Equation

摘要: 这篇论文主要研究了3D不可压Navier-Stokes方程解的H1正则性. 首先, 本论文给出井详细证明 了3D不可压Navier-Stokes方程解的局部适定性引理. 其次, 应用上述解的局部适定性引理, 第一, 可以严格证明解在小初始数据情形时的全局正则性. 第二, 对于最大可能的所有U₀和最大可能的所 有F , 证明出3D不可压Navier-Stokes方程解的H¹ 正则性. 本论文强调不仅对最大可能U0和某一 固定F 这一情形, 解具有H¹正则性, 而且对最大可能U0和最大可能的F 这一情形, 解同样具有H¹正则性。

Abstract: In this paper, we mainly study the H1 regularity of the solution of 3D incompressible Navier-Stokes equations. Firstly, the local well-fit lemma for solutions of 3D incom- pressible Navier-Stokes equations is given and proved in detail. Secondly, by applying the local well-fit lemma of the solution mentioned above, firstly, the global regularity of the solution in the case of small initial data can be proved strictly. Second, the H¹ regularity of the solution of 3D incompressible Navier-Stokes equations is proved for all possible U0 and all possible F . In this paper, it is emphasized that the solution is H¹ regular not only for the maximum possible U0 and a fixed F , but also for the maximum possible U₀ and the maximum possible F .