On the stokes operator in general unbounded domains

It is known that the Stokes operator is not well-defined in Lq-spaces for certain unbounded smooth domains unless $q=2$. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case, see {FKS1}, to the n-dimensional one, n≥ 2, replacing the space Lq, 1<∞, by Lq where Lq = Lq∩ L² for q≥ 2 and Lq = Lq+L² for 1<2. In particular, we show that the Stokes operator is well-defined in Lq for every unbounded domain of uniform C1,1-type in ⁿ, n≥ 2, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.