Abstract dissipative Hamiltonian differential-algebraic equations are everywhere

TL;DR

This paper explores how many PDE models with physical constraints can be represented as dissipative Hamiltonian differential-algebraic equations (adHDAEs), providing a unifying operator-theoretic framework for their analysis.

Contribution

It introduces a unifying operator-theoretic approach to represent and analyze PDEs as adHDAEs, highlighting their prevalence and structural properties in physical models.

Findings

Many standard PDE models can be formulated as adHDAEs.

The operator-theoretic approach unifies analysis of these PDEs.

Applications demonstrate the framework's effectiveness.

Abstract

In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there are constraints coming from the underlying physics, but many standard PDE models can be seen as an adHDAE on an extended state space. This reflects the fact that models often include closure relations and structural properties. We present a unifying operator theoretic approach to analyze the properties of such operator equations and illustrate this by several applications.