Periodic Motions of the Non-autonomous Oseen–Navier–Stokes Flows Past a Moving Obstacle with Data in Lp-Spaces

In this paper, we study the time-periodic mild solutions to the non-autonomous Oseen–Navier–Stokes equations (ONSE). We prove the existence and polynomial stability of such solutions to ONSE in the exterior domain Ω⊂ℝ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\varOmega }}\subset \mathbb {R}^{3}$\end{document} of a rigid body, D=ℝ3∖Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D=\mathbb {R}^{3}\backslash {{\varOmega }}$\end{document}, moving by time periodic motion of given period T, when the data belong to Lp-space and are sufficiently small. Our method is based on the Lp − Lq smoothness of the evolution family corresponding to linearized equations in combination with ergodic theory and fixed-point theorems to obtain the result on existence and stability of T-periodic solutions under the actions of T-periodic external forces.