Euler–Lagrange change of variables in conservation laws

Abstract

We introduce a new method for studying the Cauchy problem for systems of conservation laws in one space dimension. This method is based on the equivalence of the Cauchy problems in Eulerian and Lagrangian coordinates, as regards the existence and uniqueness of entropy solutions. The main idea is to solve the problem in Lagrangian coordinates and determine the transformation linking the two coordinates. The main contributions are the uniqueness and explicit entropy solutions. Applications include the Keyfitz–Kranzer system, the Born–Infeld (BI) equations and linear Lagrangian systems which are linear in Lagrangian coordinates. For these examples, the existence and uniqueness of solutions in L∞ are obtained in explicit expressions. The linear Lagrangian system contains examples such as the equations of pressureless gas dynamics, all 2 × 2 linearly degenerate systems and the augmented BI equations. In particular, we deduce the existence and uniqueness of entropy solutions of the Cauchy problem for the BI equations. An explicit formula of its entropy solution is also provided.