Extended Divergence-measure Fields and the Euler Equations for Gas Dynamics

A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields in Lp and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields arise naturally in the study of the behavior of entropy solutions to the Euler equations for gas dynamics and other nonlinear systems of conservation laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics.