We study distributional solutions of pressureless Euler systems on the line. In particular we show that Lagrangian solutions [7], introduced by Brenier, Gangbo, Savaré and Westdickenberg, and entropy solutions [37], studied by Nguyen and Tudorascu for the Euler–Poisson system, are equivalent. For the Euler–Poisson system this can be seen as a generalization to second-order systems of the equivalence between \(L^2\) -gradient flows and entropy solutions for a first-order aggregation equation proved by Bonaschi, Carrillo, Di Francesco and Peletier [4]. The key observation is an equivalence between Oleĭnik’s E-condition for conservation laws and a characterization due to Natile and Savaré of the normal cone for \(L^2\) -gradient flows. This new equivalence allows us to define unique solutions after blow-up for classical solutions of the Euler–Poisson system with quadratic confinement due to Carrillo, Choi and Zatorska [14], as well as to describe their asymptotic behavior.
