Well Posedness for Pressureless Flow

Abstract:

We study the uniqueness problem for pressureless gases. Previous results on this topic are only known for the case when the initial data is assumed to be a bounded measurable function. This assumption is unnatural because the solution is in general a Radon measure. In this paper, the uniqueness of the weak solution is proved for the case when the initial data is a Radon measure. We show that, besides the Oleinik entropy condition, it is also important to require the energy to be weakly continuous initially. Our uniqueness result is obtained in the same functional space as the existence theorem.