## Tohoku Mathematical Journal 2006 March SECOND SERIES VOL. 58, NO. 1

 Tohoku Math. J. 58 (2006), 129-147

Title AN $L^q$-ANALYSIS OF VISCOUS FLUID FLOW PAST A ROTATING OBSTACLE

Dedicated to Professor Hermann Sohr on his sixty-fifth birthday

Author Reinhard Farwig

(Received March 15, 2004)
Abstract. Consider the problem of time-periodic strong solutions of the Stokes and Navier-Stokes system modelling viscous incompressible fluid flow past or around a rotating obstacle in Euclidean three-space. Introducing a rotating coordinate system attached to the body, a linearization yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In this paper we find an explicit solution for the linear whole space problem when the axis of rotation is parallel to the velocity of the fluid at infinity. For the analysis of this solution in $L^q$-spaces, $1<q<\infty$, we will use tools from harmonic analysis and a special maximal operator reflecting paths of fluid particles past or around the obstacle.

2000 Mathematics Subject Classification. Primary 76D05; Secondary 35C15, 35Q35, 76D99, 76U05.

Key words and phrases. Littlewood-Paley theory, maximal operators, Oseen flow, rotating obstacles, singular integral operator, Stokes flow.