Math, Z. 196, 415-425 (1987) Mathematische Zeitschrift 9 Springer-Verlag 1987 On the Semigroup of the Stokes Operator for Exterior Domains in LO-spaces Wolfgang Borchers and Hermann Sohr Universit/it-Gesamthochschule Paderborn, Fachbereich Mathematik-Informatik, Warburger Str. 100, D-4790 Paderborn, Federal Republic of Germany 1. Introduction Let 0 ___ R" (n > 3) be an exterior domain of the n-dimensional Euclidian space. We assume that the (compact) boundary ~ g2 of f2 is of class C 2 +" with 0 < # < 1. On f2 we consider the Stokes boundary value problem (1.1) --A u+ Vp=f, div u=0, ut0a=0, for the velocity field u=(ul, u2, ...,u,)eU(~2)", l<q<o% and the pressure peUloc(O) describing a viscous incompressible flow (past a body for n=3) in the stationary linearized approximation (see [9]). Using the projection operator Pq: U(f2)"~Hq(f2) of the space Lq(I2)" onto the subspace Hq(O) of divergence-free fields with zero normal component on Of 2 (as defined in [12] or [14]), we obtain the Stokes operator Aq,=--PqA with domain D(Aq):=D(A)c~ Hq(f2). Some years ago it has been proved the important property that -Ao generates an analytic semigroup e-taq(t>O) in the space Ho(f2) (see [14], [6], [12]). However, it remained a lack of information because the boundedness [le-tAql[,<c uniformly for all t>0 could not be proved up to now for q + 2; for the Laplace operator, the corresponding property follows easily from the accretivity for l<q<oe, which does not seem to be true for the Stokes operator for q 4 = 2. The boundedness [[ e -ta" H nq < c uniformly for all t >0 is important for the investigation of the asymptotic behaviour of the velocity field u for large times t. It is the aim of the present paper to prove this boundedness property. More- c over, we prove the more general property [](2I+Aq)-ll[~<~ for all 2elE with [2[ > 0, [arg 2[ < re/2 + ~o, 0 < o9 < ~/2, from which we get further information <c for on the semigroup e -tAq. For example, we get the estimate IIAq e -tAq [In~= t all t>0, and (using [8; p. 123]) we are able to construct the fractional powers A~(0< ~< 1) of the Stokes operator also in exterior domains. Up to now this