Conservation laws and null divergences

1. Conservation laws For a system of partial differential equations, the existence of appropriate conservation laws is often a key ingredient in the investigation of its solutions and their properties. Conservation laws can be used in proving existence of solutions, decay and scattering properties, investigation of singularities, analysis of integrability properties of the system and so on. Representative applications, and more complete bibliographies on conservation laws, can be found in references [7], [8], [12], [19]. The more conservation laws known for a given system, the more tools available for the above investigations. Thus a complete classification of all conservation laws of a given system is of great interest. Not many physical systems have been subjected to such a complete analysis, but two examples can be found in [11] and [14]. The present paper arose from investigations ([15], [16]) into the conservation laws of the equations of elasticity. We begin by recalling the definition of a conservation law. Let x = (a;,..., z) be the independent and u = (it, ...,u) the dependent variables in the system. The notation #"« is an abbreviation for the collection of all with order partial derivatives of the u's with respect to the x'a, for which we use multi-index notation