We study initial-boundary value problems for the 3D Navier–Stokes equations posed on bounded and unbounded parallelepipeds as well as on bounded and unbounded smooth domains without smallness restrictions for the initial data. Under conditions on sizes of domains, we establish the existence, uniqueness and exponential decay of solutions in \(H^2\)-norm for bounded domains as well as “smoothing” effect and in \(H^1\)-norm for unbounded ones. Moreover, for smooth subdomains of unbounded domains, we prove regularity of strong solutions and “smoothing” effect.

中文翻译：

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Lipschitz和光滑域上的3D Navier-Stokes方程的整体解的指数衰减和正则性

我们研究了在有界和无界平行六面体以及有界和无界光滑域上构成的3D Navier-Stokes方程的初始边界值问题，而初始数据没有小限制。在域大小的条件下，我们建立有界域的\（H ^ 2 \）-范数解的存在性，唯一性和指数衰减以及“光滑”效应和\（H ^ 1 \）-范数的解。无限的。此外，对于无界域的平滑子域，我们证明了强解和“平滑”效应的规律性。