The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

研究了控制粘性刚性流体在刚体周围流动的方程式，该刚体以恒定的平移和旋转轴执行规定的时间周期运动。假设规定的刚体运动的周期和角速度是兼容的，并且平均平移速度不为零，则建立了时间周期解。该证明基于适当的线性化，在绝对收敛的傅立叶级数设置下进行检查。由于在经典的Sobolev空间中相应的解决问题不适当，因此在齐次Sobolev空间的框架中发展了线性理论。