On Abstract $\mathrm{grad}-\mathrm{div}$ Systems

Abstract

For a large class of dynamical problems from mathematical physics the skew-selfadjointness of a spatial operator of the form $A=\left(\begin{array}{cc}0& -C^{\ast }\\ C& 0\end{array}\right)$, where $C:D\left(C\right)\subseteq H_0\to H_1$ is a closed densely defined linear operator, is a typical property. Guided by the standard example, where $C=\mathrm{grad}=\left(\begin{array}{c}{\partial }_1\\ ⋮\\ {\partial }_n\end{array}\right)$ (and $-C^{\ast }=\mathrm{div}$, subject to suitable boundary constraints), an abstract class of operators $C=\left(\begin{array}{c}{C}_1\\ ⋮\\ C_n\end{array}\right)$ 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 $A$.