Generalized characteristics and the uniqueness of entropy solutions to zero-pressure gas dynamics

Abstract

The system of zero-pressure gas dynamics conservation laws describes the dynamics of free particles sticking under collision while mass and momentum are conserved both at the discrete and continuous levels. The existence of such solutions was established in [5]. In this paper we are concerned with the uniqueness of entropy solutions. We first introduce additionally to the Oleinik entropy condition a cohesion condition. Both conditions together form our extended concept of an admissibility condition for solutions to the system. The cohesion condition is automatically satisfied by the solutions obtained in the existence results mentioned above. Further, we regularize such a given admissible solution so that generalized characteristics are well-defined. Through limiting procedures the concept of generalized characteristics is then extended to a very large class of admissible solutions containing vacuum states and singular measures. Next we use the generalized characteristics and the dynamics of the center of mass in order to prove that all entropy solutions are equal in the sense of distributions.