Abstract. In this paper we describe two recent approaches for the L p -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle.

Consider the system of equations describing the motion of a rigid body immersed in a viscous, incompressible fluid of Newtonian or generalized Newtonian type. The class of fluids considered includes in particular shearthinning or shear-thickening fluids of power-law type of exponent d ≥ 1. We develop a method to prove that this system admits a unique, local, strong solution in the L p -setting. The approach presented in the case of generalized Newtonian fluids is based on the theory of quasi-linear evolution equations and requires that the exponent p satisfies the condition p > 5.

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We characterize the domain of the realizations of the linear parabolic operator G defined by (1.4) in L 2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L 2 regularity results for evolution equations with time-depending Ornstein-Uhlenbeck operators.2000 Mathematics Subject Classification. 47D06, 47F05, 35B65.

In this paper we investigate a class of nonautonomous linear parabolic problems with time-depending Ornstein-Uhlenbeck operators. We study the asymptotic behavior of the associated evolution operator and evolution semigroup in the periodic and non-periodic situation. Moreover, we show that the associated evolution operator is hypercontractive.2000 Mathematics Subject Classification. 47D06, 47F05, 35B65.