Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the Navier–Stokes system with the drift and (one half of the) Coriolis terms in a fixed exterior domain. The existence of the evolution operator T ( t , s ) in the space $$L^q$$ L q generated by the linearized non-autonomous system was proved by Hansel and Rhandi (J Reine Angew Math 694:1–26, 2014) and the large time behavior of T ( t , s ) f in $$L^r$$ L r for $$(t-s)\rightarrow \infty $$ ( t - s ) → ∞ was then developed by Hishida (Math Ann 372:915–949, 2018) when f is taken from $$L^q$$ L q with $$q\leqq r$$ q ≦ r . The contribution of the present paper concerns such $$L^q$$ L q - $$L^r$$ L r decay estimates of $$\nabla T(t,s)$$ ∇ T ( t , s ) with optimal rates, which must be useful for the study of stability/attainability of the Navier–Stokes flow in several physically relevant situations. Our main theorem completely recovers the $$L^q$$ L q - $$L^r$$ L r estimates for the autonomous case (Stokes and Oseen semigroups, those semigroups with rotating effect) in three dimensional exterior domains, which were established by Hishida and Shibata (Arch Ration Mech Anal 193:339–421, 2009), Iwashita (Math Ann 285, 265–288, 1989), Kobayashi and Shibata (Math Ann 310:1–45, 1998), Maremonti and Solonnikov (Ann Sc Norm Super Pisa 24:395–449, 1997) and Shibata (in: Amann, Arendt, Hieber, Neubrander, Nicaise, von Below (eds) Functional analysis and evolution equations, the Günter Lumer volume. Birkhäuser, Basel, pp 595–611, 2008).

中文翻译：

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外域中瞬态刚体运动引起的广义 Oseen 演化算子的​​梯度衰减估计

让我们考虑粘性不可压缩流体在三个维度上通过旋转刚体的运动，其中规定了物体的平移和角速度，但与时间有关。在连接到身体的参考系中，我们有 Navier-Stokes 系统，在固定的外部域中具有漂移和（一半的）科里奥利项。Hansel 和 Rhandi (J Reine Angew Math 694:1–26, 2014) 证明了进化算子 T ( t , s ) 在由线性化非自治系统生成的空间 $$L^q$$ L q 中的存在) 和 T ( t , s ) f 在 $$L^r$$ L r 中的大时间行为对于 $$(ts)\rightarrow \infty $$ ( t - s ) → ∞ 然后由 Hishida (Math Ann 372:915–949, 2018) 当 f 取自 $$L^q$$ L q 且 $$q\leqq r$$ q ≦ r 。本论文的贡献涉及这样的 $$L^q$$ L q - $$L^r$$ L r 衰减估计 $$\nabla T(t,s)$$ ∇ T ( t , s )最佳速率，这对于研究 Navier-Stokes 流在几种物理相关情况下的稳定性/可达到性必须有用。我们的主要定理在三维外部域中完全恢复了自治情况（Stokes 和 Oseen 半群，那些具有旋转效应的半群）的 $$L^q$$ L q - $$L^r$$ L r 估计，它们是由菱田和柴田 (Arch Ration Mech Anal 193:339–421, 2009)、岩下 (Math Ann 285, 265–288, 1989)、小林和柴田 (Math Ann 310:1–45, 1998 和 Solonniemon) 建立(Ann Sc Norm Super Pisa 24:395–449, 1997) 和 Shibata (in: Amann, Arendt, Hieber, Neubrander, Nicaise, von Beyond (eds) 功能分析和演化方程，Günter Lumer 卷。Birkhäuser，巴塞尔，第 595-611 页，2008 年）。