Mother Operators and their Descendants

TL;DR

This paper introduces a mechanism to generate new well-posed evolutionary equations in mathematical physics from a single spatial operator, using projections and material laws, with various examples illustrating its application.

Contribution

It presents a novel abstract method to derive a wide class of evolutionary problems from a single operator, enhancing understanding of their structure and interrelations.

Findings

Many standard evolutionary problems can be generated from one spatial operator.

The complexity of physical phenomena can be modeled through material laws.

The mechanism is demonstrated with multiple concrete examples.

Abstract

A mechanism deriving new well-posed evolutionary equations from given ones is inspected. It turns out that there is one particular spatial operator from which many of the standard evolutionary problems of mathematical physics can be generated by this abstract mechanism using suitable projections. The complexity of the dynamics of the phenomena considered can be described in terms of suitable material laws. The idea is illustrated with a number of concrete examples.