Decomposition of the homogeneous space W~(1,2) with respect to the Dirichlet form (Vu, Vv) and applications

摘要

Well known results on the generalized Stokes boundary value problem in domains Q C R~n,n≥ 2, lead to decomposition of the homogeneous space W~(1,2)(?) with respect to the Dirichlet form (Vu, Vv) in a quite natural way. On bounded Lipschitz domains this decomposition implies lower and upper bounds to the change of the Dirichlet seminorm ||Vu|| by transition from slip- to no-slip boundary condition. The bounds apply to the diffusion steps in transport-diffusion splitting schemes which are designed for numerical approximations to Navier-Stokes problems at higher Reynolds numbers. In 3 dimensions, by comparison of the Dirichlet form (Vu, Vv) with the quadratic form (curl u, curl v), from the results above we get lower and upper bounds to the change of the vorticity by transition from slip- to no-slip fluid flow on bounded Lipschitz domains.