Symmetries of Conservation Laws

We apply techniques of symmetry group analysis in solving two systems of conservation laws: a model of two strictly hyperbolic conservation laws and a zero pressure gas dynamics model, which both have no global solution, but whose solution consists of singular shock waves. We show that these shock waves are solutions in the sense of 1-strong association. Also, we compute all projectable symmetry groups and show that they are 1-strongly associated, hence transform existing solutions in the sense of 1-strong association into other solutions. The concept of classical symmetry groups offers a large number of possibilities in studying differential equations, in particular in constructing explicit solutions to linear and nonlinear differential equations or determining and classifying invariance properties [16, 17]. In various problems of mathematical physics the classical theory turns out to be insufficient, due to singular objects (like distributions or discontinuous nonlinearities) which can occur in the equation or equations with solutions in a weak sense, i.e., weak solutions (distributional, generalized or in the sense of association). Therefore, the methods of classical symmetry group analysis of differential equations have been extended to linear equations in the class of distributions [1, 2], as well as to equations involving generalized functions [5, 6, 10, 11, 9]. The aim of this paper is to apply techniques of symmetry group analysis in solving two systems of differential equations given in the form of conservation laws. The paper is divided into two parts. Section 1 provides a brief overview of the basic definitions and theorems which are going to be used for studying conservation laws. We start by recalling some facts on symmetry group analysis, which are in detail carried out in [16] (see also [17]). Then we turn to symmetries in the generalized setting, precisely to associated ones. As we will see later, the reason for this lies in the fact that the solutions of the conservation laws we consider 2000 Mathematics Subject Classification: Primary 35L65; Secondary 35D99, 46F30, 58D19.