On Abstract grad-div Systems

For a large class of dynamical problems from mathematical physics the skew-selfadjointness of a spatial operator of the form (image), where (image) is a closed densely defined linear operator between Hilbert spaces (image), is a typical property. Guided by the standard example, where (image) (and (image), subject to suitable boundary constraints), an abstract class of operators (image) is introduced (hence the title). As a particular application we consider a non-standard coupling mechanism and the incorporation of diffusive boundary conditions both modeled by setting associated with a skew-selfadjoint spatial operator A.