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.

对于在三个空间维度上的偏微分方程组，在统一框架中研究了保持在体积，曲面和曲线上的动力学守恒定律以及在曲面和曲线上保持的拓扑守恒规律。讨论了这些不同的守恒定律的全局和局部表达，包括全局运动常数的形式。主要结果包括为两个保护法则在本地或全局上是等效的，以及何时保护法在本地或全局上是微不足道的，以及推导不同类型的保护法之间的关系提供了明确的特征。特别是，对于所有类型的保护法，都明确了“琐碎的”保护法的概念。而且，作为进一步的新结果，确定了一个条件，在该条件下，一个域上的琐碎局部守恒定律可以在域边界上产生一个非平凡的全局守恒定律，并证明与包含演化方程和空间约束方程的PDE系统所具有的微分恒等式有关。来自流体流动，气体动力学，电磁学和磁流体动力学的大量物理示例被用作说明。