Sticky particles solutions for the attractive pressureless Euler-Poisson system; a projection formula and asymptotic behavior

Abstract

In this paper we perform a comprehensive study of the sticky particles solutions to the one-dimensional attractive pressureless Euler-Poisson system. We first provide a Lagrangian map characterization of the sticky particles solutions as projections onto the convex cone of essentially nondecreasing functions in $L^2(0,1)$ by following closely the approach employed earlier by Natile & Savaré for the pressureless Euler case. As a byproduct, we obtain sticky particles solutions for general initial data consisting of Borel probability measures with finite second moment and initial velocities that are square-integrable with respect to said measures. The asymptotic behavior of the sticky particles solutions is the main objective of our work; we obtain explicit exact collapse times into the equilibrium whenever such collapse occurs. In general, we prove that the sticky particles solution converges to the equilibrium in the 1-Wasserstein distance at an explicit rate.