On the different types of global and local conservation laws for partial differential equations in three spatial dimensions: review and recent developments

TL;DR

This paper reviews and advances the understanding of various global and local conservation laws in three-dimensional PDE systems, clarifying their equivalences, trivialities, and relationships with physical examples.

Contribution

It provides explicit criteria for conservation law equivalence and triviality, and explores how local trivial laws can lead to global conservation laws on boundaries.

Findings

Characterization of when two conservation laws are equivalent or trivial

Conditions under which local trivial laws generate non-trivial boundary laws

Relationships among different types of conservation laws in physical systems

Abstract

For systems of partial differential equations in three spatial dimensions, dynamical conservation laws holding on volumes, surfaces, and curves, as well as topological conservation laws holding on surfaces and curves, are studied in a unified framework. Both global and local formulations of these different conservation laws are discussed, including the forms of global constants of motion. The main results consist of providing an explicit characterization for when two conservation laws are locally or globally equivalent, and for when a conservation law is locally or globally trivial, as well as deriving relationships among the different types of conservation laws. In particular, the notion of a ``trivial'' conservation law is clarified for all of the types of conservation laws. Moreover, as further new results, conditions under which a trivial local conservation law on a domain can yield a non-trivial global conservation law on the domain boundary are determined and shown to be related to differential identities that hold for PDE systems containing both evolution equations and spatial constraint equations. Numerous physical examples from fluid flow, gas dynamics, electromagnetism, and magnetohydrodynamics are used as illustrations.