Bounded solutions of the Dirichlet problem for the Stokes resolvent system

The paper studies the Dirichlet problem for the Stokes resolvent system for bounded boundary data on bounded and unbounded domains with compact Ljapunov boundary. (The boundary might be disconnected.) For a bounded domain we prove the existence of a unique solution of the problem such that the velocity part is bounded. For an unbounded domain we prove the existence of a such solution. But this solution is not unique. We characterize all solutions of the problem. As a consequence we study bounded solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system At last we prove a generalized maximum principle for a solution of the Stokes resolvent system such that the velocity part is bounded.