Bounded solutions of the Dirichlet problem for the Stokes resolvent system

Abstract

The paper studies the Dirichlet problem for the Stokes resolvent system for bounded boundary data on bounded and unbounded domains with compact Lyapunov 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 such a solution. But this solution is not unique. We characterize all solutions of the problem. Then we study bounded solutions of the nonlinear Dirichlet problem , in , on , where F is bounded. As a consequence, we study bounded solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system , . At last we prove a generalized maximum modulus principle for a solution of the Stokes resolvent system such that the velocity part is bounded.