Abstract This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical L2-based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical L2-based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.

中文翻译：

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Navier-Stokes 方程无限能量解的全局存在性、规律性和唯一性

摘要 本文解决了与 Navier-Stokes 方程的局部能量解（在 Lemarié-Rieusset 的意义上）相关的几个问题，初始数据在使用截断的 Morrey 型量测量时在大尺度或小尺度上足够小，即： 1) 一类数据的全局存在性，包括基于 L2 的临界 Morrey 空间；(2) Navier-Stokes 方程的局部能量解的初始和最终规律，初始数据在小尺度或大尺度上足够小；(3) 基于 L2 的临界 Morrey 空间中数据的局部能量解决方案的小-大唯一性。包括许多有趣的推论，包括熟悉的 Lebesgue、Lorentz 和 Morrey 空间中的最终规律，一个新的局部广义冯·瓦尔唯一性标准，