Extended Helmholtz–Weyl decompositions

The Helmholtz–Weyl decomposition in which a vector field is decomposed into the curl of a vector potential and the gradient of a scalar potential, is extended to situations where function-objects somewhat different from vector fields are considered. This is done by creating, in the spirit of Hermann Weyl, a Hilbert space framework from which the classical as well as some new decompositions can be obtained. Because of the Hilbert space setting, functions in classes of square integrable functions are in the background. In applications to hydrodynamics, decomposition of the velocity field has to be brought into line with the decomposition of the time derivative of this field. For this purpose we introduce the derived decomposition which is what is really used in fluid mechanics. In addition there is the matter of dimensional correctness of decompositions to which attention is also paid. Applications of the theory to problems in fluid mechanics which involve dynamic boundary conditions are also given.