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

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