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The Stokes Operator with Robin Boundary Conditions in Solenoidal Subspaces of L 1(ℝ n +) and L ∞(ℝ n +)

We prove that the Stokes operator with Robin boundary conditions is the generator of a bounded holomorphic semigroup on , which is even strongly continuous on the space BUCσ(ℝ n +). Contrary to that result it is also proved that there exists no Stokes semigroup on , 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 for all types of Robin boundary conditions.