流过具有振动边界的平移体的粘性流动
Viscous flow past a translating body with oscillating boundary
摘要：
我们研究了一个不可压缩的粘性流动的障碍物周围的振荡边界，移动的平移周期运动，我们显示存在强的时间周期解的小数据在不同的配置：如果身体的平均速度为零，时间周期解的存在性内提供了一个框架内的Sobolev函数与各向同性逐点衰减。如果平均速度不为零，则可以采用该框架，但是流的空间行为需要各向异性加权空间的设置。在后一种情况下，我们也建立了一个替代的框架内的齐次索伯列夫空间的解决方案的存在性。这些结果是基于时间周期的最大正则性的相关线性化，这是来自合适的R-界的斯托克斯和Oseen预解问题。从相应的时间周期基本解推导出了点态估计。
英文摘要：
We study an incompressible viscous flow around an obstacle with an oscillating boundary that moves by a translational periodic motion, and we show existence of strong time-periodic solutions for small data in different configurations: If the mean velocity of the body is zero, existence of time-periodic solutions is provided within a framework of Sobolev functions with isotropic pointwise decay. If the mean velocity is non-zero, this framework can be adapted, but the spatial behavior of flow requires a setting of anisotropically weighted spaces. In the latter case, we also establish existence of solutions within an alternative framework of homogeneous Sobolev spaces. These results are based on the time-periodic maximal regularity of the associated linearizations, which is derived from suitable R-bounds for the Stokes and Oseen resolvent problems. The pointwise estimates are deduced from the associated time-periodic fundamental solutions.
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