The Stokes operator with Robin boundary conditions in solenoidal subspaces of L 1(R n +) and L ∞(R n +)
Juergen Saal · Hokkaido University Collection of Scholarly and Academic Papers (Hokkaido University) · 2004
We prove that the Stokes operator with Robin boundary conditions is the generator of a bounded holomorphic semigroup on L^\\infty_\\sigma({\\mathbb R}^n_+), which is even strongly continuous on the space \\BUC_\\sigma({\\mathbb R}^n_+). Contrary to that result it is also proved that there exists no Stokes semigroup on L^1_\\sigma({\\mathbb R}^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^1_\\sigma({\\mathbb R}^n_+) for all types of Robin boundary conditions.