We study global regularity properties of invariant measures associated with second order differential operators in R N . Under suitable conditions, we prove global boundedness of the density, Sobolev regularity, a Harnack inequality and pointwise upper and lower bounds.

Consider the incompressible Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity -sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to a non-autonomous problem with unbounded drift terms on a fixed exterior domain R d . It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system ¹T .t; s/º t s 0 on L p . / for 1 < p < 1. Moreover, L p -L q smoothing properties and gradient estimates of T .t; s/, 0 Ä s Ä t, are obtained. These results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for initial value in L p . /, p d .

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