Pressureless Euler/Euler-Poisson Systems via Adhesion Dynamics and Scalar Conservation Laws

The “sticky particles” model at the discrete level is employed to obtain global solutions for a class of systems of conservation laws among which lie the pressureless Euler and the pressureless attractive/repulsive Euler-Poisson system with zero background charge. We consider the case of finite, nonnegative initial Borel measures with finite second-order moment, along with continuous initial velocities of at most quadratic growth and finite energy. We prove the time regularity of the solution for the pressureless Euler system and obtain that the velocity satisfies the Oleinik entropy condition, which leads to a partial result on uniqueness. Our approach is motivated by earlier work of Brenier and Grenier who showed that one dimensional conservation laws with special initial conditions and fluxes are appropriate for studying the pressureless Euler system.