Abstract

Consider the system of Navier–Stokes equations in a perturbed half-space. We study the existence and stability of periodic solutions and almost periodic solutions to this system. For the Stokes system we use \(L^p\text {-}L^q\) estimates in combination with interpolation spaces to show the existence of bounded (in time) mild solutions as well as the existence of periodic solutions. Moreover, the existence of almost periodic solutions is proved. Then the existence, uniqueness and stability of the periodic and almost periodic solutions to the Navier–Stokes system are proved by using fixed-point arguments and certain interpolation relations.

中文翻译：

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纳维-斯托克斯流的周期性和准周期性运动及其在扰动半空间中的稳定性

摘要

考虑扰动半空间中的纳维-斯托克斯方程组。我们研究该系统的周期解和近似周期解的存在性和稳定性。对于斯托克斯系统，我们使用\(L^p\text {-}L^q\)估计与插值空间相结合来显示有界（时间上）温和解的存在以及周期解的存在。此外，还证明了近似周期解的存在性。然后利用不动点参数和一定的插值关系证明了纳维-斯托克斯系统周期解和准周期解的存在性、唯一性和稳定性。