The Stokes operator with Robin boundary conditions in solenoidal subspaces of L1(Rn+) and L∞(Rn+)

We prove that the Stokes operator with Robin boundary conditions is the generator of a bounded holomorphic semigroup on L-sigma(infinity)(IR+n), which is even strongly continuous on the space BUC sigma(IR+n). Contrary to that result it is also proved that there exists no Stokes semigroup on L-sigma(1)(IR+n), except if we assume the special case of Neumann boundary conditions. Nevertheless, we also obtain gradient estimates for the solution of the Stokes equations in L-sigma(1)(IR+n), for all types of Robin boundary conditions.