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

TL;DR

This paper investigates the skew-selfadjointness of a broad class of abstract differential operators, extending the classical grad-div framework, with applications to non-standard couplings and boundary conditions in mathematical physics.

Contribution

It introduces a generalized class of operators combining multiple differential components, expanding the theoretical understanding of skew-selfadjoint operators in physics models.

Findings

Established conditions for skew-selfadjointness of the generalized operators

Applied the framework to non-standard coupling mechanisms

Incorporated diffusive boundary conditions into the operator model

Abstract

For a large class of dynamical problems from mathematical physics the skew-selfadjointness of a spatial operator of the form $A=\left(\begin{array}{cc} 0 & -C^{*}\\ C & 0 \end{array}\right)$, 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 $C=\mathrm{grad}=\left(\begin{array}{c} \partial_{1}\\ \vdots\\ \partial_{n} \end{array}\right)$ (and $-C^{*}=\mathrm{div}$, subject to suitable boundary constraints), an abstract class of operators $C=\left(\begin{array}{c} C_{1}\\ \vdots\\ C_{n} \end{array}\right)$ 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 $A$.