In this paper we consider classes of initial data that ensure local-in-time Hadamard well-posedness of the associated weak Leray–Hopf solutions of the three-dimensional Navier–Stokes equations. In particular, for any solenodial $$L_{2}$$ initial data $$u_{0}$$ belonging to certain subsets of $$VMO^{-1}(\mathbb {R}^3)$$ , we show that weak Leray–Hopf solutions depend continuously with respect to small divergence-free $$L_{2}$$ perturbations of the initial data $$u_{0}$$ (on some finite-time interval). Our main result is inspired by and improves upon previous work of the author (Barker in J Math Fluid Mech 20(1):133–160, 2018) and work of Jean–Yves Chemin (Commun Pure Appl Math 64(12):1587–1598, 2011). Our method builds upon [4, 9]. In particular our method hinges on decomposition results for the initial data inspired by Calderon (Trans Am Math Soc 318(1):179–200, 1990) together with use of persistence of regularity results. The persistence of regularity statement presented may be of independent interest, since it does not rely upon the solution or the initial data being in the perturbative regime.

中文翻译：

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关于 Navier-Stokes 方程能量解的 $$L_{2}$$ 初始数据的局部连续性

在本文中，我们考虑初始数据的类别，以确保三维 Navier-Stokes 方程的相关弱 Leray-Hopf 解的局部时间 Hadamard 适定性。特别地，对于属于 $$VMO^{-1}(\mathbb {R}^3)$$ 某些子集的任何 solenodial $$L_{2}$$ 初始数据 $$u_{0}$$ ，我们表明弱 Leray-Hopf 解决方案持续依赖于初始数据 $$u_{0}$$（在某个有限时间间隔内）的小无散度 $$L_{2}$$ 扰动。我们的主要结果受到作者先前工作的启发和改进（Barker in J Math Fluid Mech 20(1):133–160, 2018）和 Jean-Yves Chemin (Commun Pure Appl Math 64(12):1587 –1598, 2011)。我们的方法建立在 [4, 9] 之上。特别是，我们的方法取决于受到 Calderon（Trans Am Math Soc 318(1):179–200, 1990）启发的初始数据的分解结果以及规律结果的持久性的使用。所呈现的规律性陈述的持久性可能具有独立意义，因为它不依赖于处于微扰状态的解或初始数据。