Maximal Regularity of the Stokes Operator in General Unbounded Domains of ℝ n

Abstract

It is well known that the Helmholtz decomposition of L q -spaces fails to exist for certain unbounded smooth domains unless q ≠ 2. Hence also the Stokes operator and the Stokes semigroup are not well defined for these domains when q ≠ 2. In this note, we generalize a new approach to the Stokes operator in general unbounded domains from the three-dimensional case, see [], to the n-dimensional one, n ≥ 2, by replacing the space L q , 1 < q < ∞, by \( \tilde L^q {\text{ where }}\tilde L^q \) = L q ∩ L 2 for q ≥ 2 and \( \tilde L_q \) = L q + L 2 for 1 < q < 2. As a main result we show that the nonstationary Stokes equation has maximal regularity in L 8(0, T; \( \tilde L_q \) ), 1 < s, q < ∞, T > 0, for every unbounded domain of uniform C 1,1-type in ℝ n .