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 A=(0−C⁎C0), where C:D(C)⊆H0→H1C:D(C)⊆H0→H1 is a closed densely defined linear operator between Hilbert spaces H0,H1H0,H1, is a typical property. Guided by the standard example, where C=grad=(∂1⋮∂n) (and −C⁎=div−C⁎=div, subject to suitable boundary constraints), an abstract class of operators C=(C1⋮Cn) 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.