Weak Diffeomorphisms and Extremals for Scalar Conservation Laws

Scalar conservation laws in one space variable allow a Lagrangian (particle path) formulation. The Lagrangian trajectory in the infinite-dimensional group of diffeomorphisms on the physical space can be written as a system of conservation laws. The relation between solutions of the Cauchy problem for the conservation law and solutions of the corresponding Cauchy problem on the diffeomorphism group extends to weak solutions of the coresponding problems. The correspondence between particle paths and transport equations is analogous to that between a Lie group and the corresponding Lie algebra. This paper establishes that for scalar conservation laws the particle paths are extremals of an action functional on the space of diffeomorphisms; that is, they are geodesics in some metric. In some examples of systems of conservation laws, including the physical example of isentropic gas dynamics in one space dimension, diffeomorphism representations also exist and may be interpreted as extremals of action functionals.